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class="fa-fw fa-brands fa-cloudsmith"></i><span> 概率图学习</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%9D%9E%E5%8F%82%E6%95%B0%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-codepen"></i><span> 非参数贝叶斯模型</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E8%A1%A8%E7%A4%BA%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-solid fa-cube"></i><span> 表示学习</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E5%8F%AF%E8%A7%A3%E9%87%8A%E6%80%A7/"><i class="fa-fw fa-solid fa-ghost"></i><span> 可解释性</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%99%8D%E7%BB%B4/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 降维</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E8%81%9A%E7%B1%BB/"><i class="fa-fw fa-solid 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href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E5%8F%98%E5%88%86%E6%8E%A8%E6%96%AD/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 变分推断</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%BF%91%E4%BC%BC%E8%B4%9D%E5%8F%B6%E6%96%AF%E8%AE%A1%E7%AE%97/"><i class="fa-fw fa-solid fa-cube"></i><span> 近似贝叶斯计算</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%A8%A1%E5%9E%8B%E6%AF%94%E8%BE%83%E4%B8%8E%E9%80%89%E6%8B%A9/"><i class="fa-fw fa-solid fa-ghost"></i><span> 模型比较与选择</span></a></li><li><a class="site-page child" href="/categories/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BB%9F%E8%AE%A1/%E8%B4%9D%E5%8F%B6%E6%96%AF%E4%BC%98%E5%8C%96/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 贝叶斯优化</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-ghost"></i><span> 不确定性DL</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/BayesNN/%E6%A6%82%E8%A7%88"><i class="fa-fw fa-solid fa-cube"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E5%8D%95%E4%B8%80%E7%A1%AE%E5%AE%9A%E6%80%A7%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 单一确定性神经网络</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-brands fa-deezer"></i><span> 贝叶斯神经网络</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E6%B7%B1%E5%BA%A6%E9%9B%86%E6%88%90/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 深度集成</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E6%95%B0%E6%8D%AE%E5%A2%9E%E5%BC%BA/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 数据增强</span></a></li><li><a class="site-page child" href="/categories/BayesNN/%E5%AF%B9%E6%AF%94%E4%B8%8E%E8%AF%84%E6%B5%8B/"><i class="fa-fw fa-brands fa-deezer"></i><span> 对比与评测</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-map"></i><span> 空间统计</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/GeoAI/%E7%BB%BC%E8%BF%B0%E7%B1%BB/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%82%B9%E5%8F%82%E8%80%83%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-solid fa-map"></i><span> 点参考数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E9%9D%A2%E5%85%83%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 面元数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%82%B9%E6%A8%A1%E5%BC%8F%E6%95%B0%E6%8D%AE/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 点模式数据</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E8%B4%9D%E5%8F%B6%E6%96%AF%E6%96%B9%E6%B3%95/"><i class="fa-fw fa-solid fa-cube"></i><span> 空间贝叶斯方法</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E5%8F%98%E7%B3%BB%E6%95%B0%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-ghost"></i><span> 空间变系数模型</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E7%A9%BA%E9%97%B4%E7%BB%9F%E8%AE%A1%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-brands fa-deezer"></i><span> 空间统计深度学习</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E6%97%B6%E7%A9%BA%E7%BB%9F%E8%AE%A1%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-atlas"></i><span> 时空统计模型</span></a></li><li><a class="site-page child" href="/categories/GeoAI/%E5%A4%A7%E6%95%B0%E6%8D%AE%E4%B8%93%E9%A2%98/"><i class="fa-fw fa fa-anchor"></i><span> 大数据专题</span></a></li><li><a class="site-page child" href="/categories/GeoAI/GeoAI/"><i class="fa-fw fa-brands fa-codepen"></i><span> GeoAI</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-database"></i><span> 基础</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 高等数学</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E6%A6%82%E7%8E%87%E4%B8%8E%E7%BB%9F%E8%AE%A1/"><i class="fa-fw fa-brands fa-deezer"></i><span> 概率与统计</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E7%BA%BF%E4%BB%A3%E4%B8%8E%E7%9F%A9%E9%98%B5%E8%AE%BA/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 线代与矩阵论</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E6%9C%80%E4%BC%98%E5%8C%96%E7%90%86%E8%AE%BA/"><i class="fa-fw fa-brands fa-codepen"></i><span> 最优化理论</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E4%BF%A1%E6%81%AF%E8%AE%BA/"><i class="fa-fw fa-solid fa-cube"></i><span> 信息论</span></a></li><li><a class="site-page child" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0%E6%A8%A1%E5%9E%8B/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-ghost"></i><span> 机器学习</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E7%9F%A5%E8%AF%86%E5%9B%BE%E8%B0%B1/"><i class="fa-fw fa-solid fa-globe"></i><span> 知识图谱</span></a></li><li><a class="site-page child" href="/categories/%E5%9F%BA%E7%A1%80%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/%E8%87%AA%E7%84%B6%E8%AF%AD%E8%A8%80%E5%A4%84%E7%90%86/"><i class="fa-fw fa-solid 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href="https://xishansnow.github.io/ElementsOfStatisticalLearning/index.html"><i class="fa-fw fa-solid  fa-book-atlas"></i><span> 《统计学习精要（ESL）》</span></a></li><li><a class="site-page child" href="https://xishansnow.github.io/spatialSTAT_CN/index.html"><i class="fa-fw fa-solid  fa-layer-group"></i><span> 《空间统计学》</span></a></li><li><a class="site-page child" target="_blank" rel="noopener" href="https://otexts.com/fppcn/index.html"><i class="fa-fw fa-solid  fa-cloud-sun-rain"></i><span> 《预测：方法与实践》</span></a></li><li><a class="site-page child" href="https://xishansnow.github.io/MLAPP/index.html"><i class="fa-fw fa-solid  fa-robot"></i><span> 《机器学习的概率视角（MLAPP）》</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-compass"></i><span> 索引</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/archives/"><i class="fa-fw fa-solid fa-timeline"></i><span> 时间索引</span></a></li><li><a class="site-page child" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签索引</span></a></li><li><a class="site-page child" href="/categories/"><i class="fa-fw fas fa-folder-open"></i><span> 分类索引</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-link"></i><span> 其他</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/link/food/"><i class="fa-fw fas fa-utensils"></i><span> 美食博主</span></a></li><li><a class="site-page child" href="/link/photography"><i class="fa-fw fas fa-camera"></i><span> 摄影大神</span></a></li><li><a class="site-page child" href="/link/paper/"><i class="fa-fw fas fa-book-open"></i><span> 学术工具</span></a></li><li><a class="site-page child" href="/gallery/"><i class="fa-fw fas fa-images"></i><span> 摄影作品</span></a></li><li><a class="site-page child" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 关于</span></a></li></ul></div></div></div></div><div class="post" id="body-wrap"><header class="post-bg" id="page-header" style="background-image: url('/img/coffe_03.png')"><nav id="nav"><span id="blog_name"><a id="site-name" href="/">西山晴雪的知识笔记</a></span><div id="menus"><div id="search-button"><a class="site-page social-icon search"><i class="fas fa-search fa-fw"></i><span> 搜索</span></a></div><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页</span></a></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-atom"></i><span> 预测</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E5%B9%BF%E4%B9%89%E7%BA%BF%E6%80%A7%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-atom"></i><span> 广义线性模型</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E9%9D%9E%E5%8F%82%E6%95%B0%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fas fa-cogs"></i><span> 传统非参数模型</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E9%AB%98%E6%96%AF%E8%BF%87%E7%A8%8B/"><i class="fa-fw fas fa-school"></i><span> 高斯过程</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C/"><i class="fa-fw fas fa-layer-group"></i><span> 神经网络</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E6%A8%A1%E5%9E%8B%E9%80%89%E6%8B%A9%E4%B8%8E%E5%B9%B3%E5%9D%87/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 模型选择与平均</span></a></li><li><a class="site-page child" href="/categories/%E9%A2%84%E6%B5%8B%E4%BB%BB%E5%8A%A1/%E5%B0%8F%E6%A0%B7%E6%9C%AC%E5%AD%A6%E4%B9%A0/"><i class="fa-fw fa-solid fa-globe"></i><span> 小样本学习</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-file-export"></i><span> 生成</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E4%BC%A0%E7%BB%9F%E6%A6%82%E7%8E%87%E5%9B%BE%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 传统概率图模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA/"><i class="fa-fw fa-solid fa-deezer"></i><span> 玻耳兹曼机</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E5%8F%98%E5%88%86%E8%87%AA%E7%BC%96%E7%A0%81%E5%99%A8/"><i class="fa-fw fa-brands fa-cloudsmith"></i><span> 变分自编码器</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E8%87%AA%E5%9B%9E%E5%BD%92%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-brands fa-codepen"></i><span> 自回归模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E5%BD%92%E4%B8%80%E5%8C%96%E6%B5%81/"><i class="fa-fw fa-solid fa-cube"></i><span> 归一化流</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E6%89%A9%E6%95%A3%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-ghost"></i><span> 扩散模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E8%83%BD%E9%87%8F%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-gas-pump"></i><span> 能量模型</span></a></li><li><a class="site-page child" href="/categories/%E7%94%9F%E6%88%90%E4%BB%BB%E5%8A%A1/%E7%94%9F%E6%88%90%E5%BC%8F%E5%AF%B9%E6%8A%97%E7%BD%91%E7%BB%9C/"><i class="fa-fw fa-solid fa-globe"></i><span> 生成式对抗网络</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fas fa-magnet"></i><span> 挖掘</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E6%A6%82%E8%A7%88/"><i class="fa-fw fa-solid fa-hands-holding"></i><span> 概览</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E9%9A%90%E5%9B%A0%E5%AD%90%E6%A8%A1%E5%9E%8B/"><i class="fa-fw fa-solid fa-chart-area"></i><span> 隐因子模型</span></a></li><li><a class="site-page child" href="/categories/%E5%8F%91%E7%8E%B0%E4%BB%BB%E5%8A%A1/%E7%8A%B6%E6%80%81%E7%A9%BA%E9%97%B4%E6%A8%A1%E5%9E%8B/"><i class="fa-fw 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<link rel="stylesheet" type="text&#x2F;css" href="https://cdn.jsdelivr.net/hint.css/2.4.1/hint.min.css"><p>【摘 要】  时空数据在农业、生态和环境科学中无处不在，研究它们对于理解和预测各种过程非常重要。对随时间变化的空间过程建模的困难之一是必须描述这种过程如何变化的依赖结构的复杂性，以及高维复杂数据集和大型预测域的存在。为非线性动态时空模型 (DSTM) 指定参数化尤其具有挑战性，这些模型在科学上和计算上都非常有用。统计学家开发了深层分层模型，可以适应过程的复杂性以及预测和推断中的不确定性。然而，这些模型可能很昂贵并且通常是特定于应用程序的。另一方面，机器学习社区已经为非线性时空建模开发了替代的“深度学习”方法。这些模型很灵活，但通常不会在概率框架中实现。这两种范式有许多共同点，并提出了可以从每个框架的元素中受益的混合方法。这篇概述论文简要介绍了深度分层 DSTM (DH-DSTM) 框架和机器学习中的深度模型，最后介绍了深度神经网络动态时空模型 (DN-DSTM)，将来自 DH-DSTM 和 DN-DSTM 的要素结合起来的最新方法作为插图呈现。</p>
<p>【引 文】 C. K. Wikle, “Comparison of Deep Neural Networks and Deep Hierarchical Models for Spatio-Temporal Data.” arXiv, Feb. 21, 2019. Accessed: Nov. 15, 2022. [Online]. Available: <a target="_blank" rel="noopener" href="http://arxiv.org/abs/1902.08321">http://arxiv.org/abs/1902.08321</a></p>
<h2 id="1-简介">1 简介</h2>
<p>深度学习是一种机器学习 (ML)，它利用一组相互连接的分层模型来预测或分类复杂数据集的元素。机器学习的深度学习革命是相对较新的，主要与神经模型相关，例如前馈神经网络 (FNN)、卷积神经网络 (CNN)、递归神经网络 (RNN)、生成对抗网络 (GAN)，或这些神经网络的某种组合架构。有与这些方法相关的非凡成功案例，例如可以击败围棋、国际象棋或将棋专家的模型（Silver 等人，2016 年、2018 年），当然也有失败的案例（Shalev-Shwartz 等人., 2017)，尽管宣传较少。统计学家不应该对这些深度机器学习方法的成功（和失败）感到惊讶，因为我们多年来一直也在使用深度层次模型 (Deep Hierarchical Models, HM)。</p>
<p>深层层次模型是指多级的贝叶斯分层模型。事实上，深度机器学习和深度分层模型成功和失败的许多原因是相同的。本文的主要目的是在时空建模背景下讨论其中的一些联系，并展示一些可以在传统统计建模框架内利用深度机器学习模型的方法。</p>
<p>时空过程在环境科学中无处不在。他们描述了空间相关的过程如何随着时间的推移而变化，并受到各种驱动机制的影响。此类过程的一个重要建模挑战涉及如何解释不同尺度空间和时间变异性之间的相互作用、感兴趣过程内部以及过程之间（内外生）的相互作用。时空过程在时间上是通常完全非线性，至少在某些时间或空间尺度上是这样。虽然在这种情况下已经有了一些参数时空统计模型，但大多要结合感兴趣系统的潜在动力学知识（例如 Wikle 等人，2001 年；Wikle 和 Hooten，2010 年）。这种 <code>深度分层动态时空模型 (DH-DSTM)</code> 可能非常复杂。同样，深度机器学习方法中最成功的案例可能与具有复杂空间和时间依赖性的数据相关联。特别是，CNN 模型在视觉和图像处理方面非常成功，而 RNN 模型利用了语言处理中复杂的时间依赖性（参见 Goodfellow 等的概述，2016 年；Aggarwal，2018 年）。 CNN 和 RNN 方法越来越多地结合起来对时空过程进行建模（例如，Donahue 等人，2015 年）。在本文中，我们将这种混合时空模型称为 <code>深度神经动力学时空模型 (DN-DSTM)</code>。</p>
<p>面对复杂的时空建模挑战，环境统计学家该如何决定哪种范式最适合他们的问题呢？ DH-DSTM 和 DN-DSTM 的实施都具有挑战性，通常需要大量训练数据和专门的计算算法。正如第 4.6 节中所讨论的，这两种建模范式对这些挑战共享共同的或相似的解决方案。</p>
<p>还必须考虑不确定性量化 (UQ) 对手头问题的重要性。作为统计学家，我们认为不确定性量化对事情始终具有根本的重要性，但现实情况是，在某些情况下，人们只需要进行预测或分类，而不确定性量化只是次要的。大多数 DN-DSTM 方法不提供基于模型的不确定性度量，而 DH-DSTM 方法建立在一个框架之上，以明确捕获问题多方面（数据、过程和参数）的不确定性。但是，DN-DSTM 模型确实可以灵活地及时考虑非马尔可夫反馈机制和遥远过去特定事件的影响，而 DH-DSTM 通常基于马尔可夫（即自回归）结构。这表明我们有机会借鉴 DH-DSTM 和 DN-DSTM 方法的想法来开发相对简约和灵活的模型，这些模型可以以计算上可行的方式适应现实世界的复杂性和不确定性量化。也许更重要的是，在某些情况下，这些方法可用于无法访问大量数据源（标记或未标记）的情况，尤其是当其与简约架构链接在一起时。</p>
<p>本文机构如下：</p>
<ul>
<li>第 2 节从描述性和动态性角度简要概述了统计中的时空建模，说明了基函数表示的重要性。</li>
<li>第 3 节简要概述了深度建模和 DH-DSTM 统计观点。</li>
<li>第 4 节简要概述了机器学习中的深度模型以及与其实现相关的问题，包括深度前馈神经网络 (DNN)、CNN 、RNN 和 DN-DSTM。</li>
<li>第 5 节回顾了一些最近用于链接 DH-DSTM 和 DNDSTM 框架的方法。</li>
<li>第 6 节介绍了结论性讨论。</li>
</ul>
<h2 id="2-时空建模概述">2 时空建模概述</h2>
<p>在统计学中，我们通常对遵循观测模型和时空隐过程模型的一般形式的时空模型感兴趣（例如 Cressie 和 Wikle，2011 年；Wikle 等人，2019 年）：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mo stretchy="false">[</mo><mtext>observations | latent process and obs/sampling error</mtext><mo stretchy="false">]</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(1)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">[\text{observations | latent process and obs/sampling error}] \tag{1}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord text"><span class="mord">observations | latent process and obs/sampling error</span></span><span class="mclose">]</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtext>latent process = “fixed effects” + dependent random process</mtext></mtd><mtd width="50%"></mtd><mtd><mtext>(2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\text{latent process = “fixed effects” + dependent random process} \tag{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">latent process = “fixed effects” + dependent random process</span></span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中符号 “ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\cdot ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">⋅</span><span class="mclose">]</span></span></span></span> ” 表示通用分布，符号 “ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\mid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∣</span></span></span></span> ” 表示条件，模型的每个组件都在空间和时间上进行索引。</p>
<p>更具体地说，假设我们对潜在（未观测到的）时空过程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="bold">s</mi><mo>∈</mo><msub><mi>D</mi><mi>s</mi></msub><mo separator="true">,</mo><mi>t</mi><mo>∈</mo><msub><mi>D</mi><mi>t</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Y (\mathbf{s};t) : \mathbf{s} \in  D_s, t \in  D_t\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathbf">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 感兴趣，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">s</mi></mrow><annotation encoding="application/x-tex">\mathbf{s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">s</span></span></span></span> 是域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">D_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> 维实空间的子集）中的空间位置， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 是时间域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">D_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 中的时间索引（沿一维实线）。则对空间位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi mathvariant="bold">s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>m</mi><mi>j</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\mathbf{s}_{ij} : i = 1, \ldots , m_j\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 和时间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>t</mi><mi>j</mi></msub><mo>:</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>T</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t_j : j = 1, \ldots , T \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">}</span></span></span></span>。 <code>式（1）</code> 中一个常见的高斯时空观测示例由下式给出：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>z</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">;</mo><msub><mi>t</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">;</mo><msub><mi>t</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">z(\mathbf{s}_{ij}; t_j) = Y (\mathbf{s}_{ij}; t) + \epsilon(\mathbf{s}_{ij}; t_j) \tag{3}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal">ϵ</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">;</mo><msub><mi>t</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo><mover><mo><mo>∼</mo></mo><mrow><mi>i</mi><mi mathvariant="normal">.</mi><mi>i</mi><mi mathvariant="normal">.</mi><mi>d</mi></mrow></mover></mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{s}_{ij}; t_j) \stackrel{i.i.d} \sim \text{Gau}(0, \sigma^2 )</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4391em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">∼</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">.</span><span class="mord mathnormal mtight">i</span><span class="mord mtight">.</span><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是观测误差过程。<code>式（2）</code> 中的隐高斯时空过程可以表示为：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>η</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">Y (\mathbf{s};t) = \mu (\mathbf{s};t) + η(\mathbf{s};t) \tag{4}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu (\mathbf{s};t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 是时空均值函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">η(\mathbf{s};t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 是具有协方差函数的零均值 <code>高斯过程 (Gaussian Process, GP)</code>，比如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>η</mi></msub><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>η</mi><mo stretchy="false">(</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><msup><mi>t</mi><mo mathvariant="normal">′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mtext>Cov</mtext><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>η</mi><mo stretchy="false">(</mo><msup><mi mathvariant="bold">s</mi><mo mathvariant="normal">′</mo></msup><mo separator="true">;</mo><msup><mi>t</mi><mo mathvariant="normal">′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_η(η(\mathbf{s};t), η(s&#x27;, t^\prime)) ≡ \text{Cov}(η(\mathbf{s};t), η(\mathbf{s}^\prime;t^\prime))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.038em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Cov</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span>。那么，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y(\mathbf{s};t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 就是一个具有均值函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu (\mathbf{s};t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 和协方差函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>η</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_η(·,·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅,⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 的GP。</p>
<p>回想一下，高斯过程是关于函数的分布，它完全由在感兴趣的时空域（例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>s</mi></msub><mo>×</mo><msub><mi>D</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">D_s × D_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ）上定义的均值函数和协方差函数指定。 高斯过程具有非常有用的属性：其任意有限维分布都是高斯分布（即正态分布）。</p>
<p>现在，假设我们有兴趣在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><msub><mo>∑</mo><mi>j</mi></msub><msub><mi>m</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">m = \sum_j m_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1858em;vertical-align:-0.4358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 维观测向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>≡</mo><mo stretchy="false">{</mo><mi>z</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">;</mo><msub><mi>t</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">z ≡ \{z(\mathbf{s}_{ij}; t_j)\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)}</span></span></span></span> 的情况下预测位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{s}_0; t_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的隐过程。时空（通用）克里金法是一个最优线性预测 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>Y</mi><mo>^</mo></mover><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat{Y} (\mathbf{s}_0; t_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，它通过最小化均方预测误差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mover accent="true"><mi>Y</mi><mo>^</mo></mover><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E(Y (\mathbf{s}_0; t_0) −\hat{Y} (\mathbf{s}_0; t_0))^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 获得：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mover accent="true"><mi>Y</mi><mo>^</mo></mover><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold">x</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mo mathvariant="normal">′</mo></msup><msub><mover accent="true"><mi mathvariant="bold-italic">β</mi><mo>^</mo></mover><mtext>gls</mtext></msub><mo>+</mo><msubsup><mi mathvariant="bold">c</mi><mn>0</mn><mo mathvariant="normal">′</mo></msubsup><msubsup><mi mathvariant="bold">C</mi><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold">z</mi><mo>−</mo><mi mathvariant="bold">X</mi><msub><mover accent="true"><mi mathvariant="bold-italic">β</mi><mo>^</mo></mover><mtext>gls</mtext></msub><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(5)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\hat{Y}(\mathbf{s}_0; t_0) = \mathbf{x}(\mathbf{s}_0; t_0)^\prime \hat{\boldsymbol{\beta}}_{\text{gls}} + \mathbf{c}^\prime_0 \mathbf{C}^{−1}_z (\mathbf{z} − \mathbf{X} \hat{\boldsymbol{\beta}}_{\text{gls}}) \tag{5}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3381em;vertical-align:-0.3802em;"></span><span class="mord mathbf">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">gls</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.3381em;vertical-align:-0.3802em;"></span><span class="mord mathbf">X</span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">gls</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1.3381em;vertical-align:-0.3802em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">x</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{x}(\mathbf{s}_0; t_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是待估计位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{s}_0; t_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 处的已知协变量，是一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 维向量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">β</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\beta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span></span></span></span> 是对应的参数向量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">X</span></span></span></span> 是所有已观测位置处的协变量构成的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">m × p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 矩阵，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">C</mi><mi>z</mi></msub><mo>≡</mo><mi mathvariant="normal">Var</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{C}_z ≡ \operatorname{Var}(\mathbf{z})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">Var</span></span><span class="mopen">(</span><span class="mord mathbf">z</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">m × m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 协方差矩阵，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">c</mi><mn>0</mn></msub><mo>≡</mo><msub><mi>c</mi><mi>η</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">z</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{c}_0 ≡ c_η(\mathbf{z}, Y(\mathbf{s}_0; t_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6138em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是观测位置和待预测位置之间的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">m × 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 协方差向量，<code>式(5)</code> 中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">β</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\beta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span></span></span></span> 广义最小二乘估计由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi mathvariant="bold-italic">β</mi><mo>^</mo></mover><mtext>gls</mtext></msub><mo>≡</mo><mo stretchy="false">(</mo><msup><mi mathvariant="bold">X</mi><mo mathvariant="normal">′</mo></msup><msubsup><mi mathvariant="bold">C</mi><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mi mathvariant="bold">X</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="bold">X</mi><mo mathvariant="normal">′</mo></msup><msubsup><mi mathvariant="bold">C</mi><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\hat{\boldsymbol{\beta}}_{\text{gls}} ≡ (\mathbf{X}^\prime\mathbf{C}^{−1}_z \mathbf{X})^{−1} \mathbf{X}^\prime\mathbf{C}^{-1}_z \mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3381em;vertical-align:-0.3802em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">gls</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathbf">X</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathbf">z</span></span></span></span> 给出。注意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">C</mi><mi>z</mi></msub><mo>=</mo><msub><mi mathvariant="bold">C</mi><mi>y</mi></msub><mo>+</mo><msubsup><mi>σ</mi><mi>ϵ</mi><mn>2</mn></msubsup><mi mathvariant="bold">I</mi><mo>=</mo><msub><mi mathvariant="bold">C</mi><mi>η</mi></msub><mo>+</mo><msubsup><mi>σ</mi><mi>ϵ</mi><mn>2</mn></msubsup><mi mathvariant="bold">I</mi></mrow><annotation encoding="application/x-tex">\mathbf{C}_z = \mathbf{C}_y + \sigma^2_\epsilon \mathbf{I} = \mathbf{C}_{\eta} + \sigma^2_\epsilon \mathbf{I}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathbf">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">ϵ</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathbf">I</span></span></span></span>。</p>
<p>相关的时空克里金法方差由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mi>Y</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>c</mi><mrow><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mo>−</mo><msubsup><mi mathvariant="bold">c</mi><mn>0</mn><mo mathvariant="normal">′</mo></msubsup><msubsup><mi mathvariant="bold">C</mi><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi mathvariant="bold">c</mi><mn>0</mn></msub><mo>+</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\sigma^2_Y (\mathbf{s}_0; t_0) = c_{0,0} − \mathbf{c}^\prime_0 \mathbf{C}^{-1}_z \mathbf{c}_0 + κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0894em;vertical-align:-0.2753em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4247em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2753em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathbf">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 给出， 其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mrow><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mo>≡</mo><mi mathvariant="normal">Var</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">s</mi><mn>0</mn></msub><mo separator="true">;</mo><msub><mi>t</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_{0,0} ≡ \operatorname{Var}(Y (\mathbf{s}_0; t_0))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7499em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">Var</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">κ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> 表示由于估计 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">β</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\beta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span></span></span></span> 而给预测带来的不确定性（例如 Wikle 等人，2019）。直接修改这些公式以获得多个位置的预测是直接的，并且该方法也可以扩展到非高斯数据模型，但k可能没有封闭形式解（例如，参见 Cressie 和 Wikle，2011）</p>
<p>这种时空建模方法是描述性的，因为它只依赖于隐过程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Y(\mathbf{s};t)\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)}</span></span></span></span> 的一阶矩和二阶矩。这在人们对底层的时空过程不太了解非常有用，此时只需要按照类似于 “地理学第一定律”（Tobler，1970） 的指导，指定一个合理的时空协方差结构（和时空趋势）就能构造可工作的模型。不过，这对于复杂过程来说可能具有挑战性，因为在 Tobler 定律可能不成立的许多情况下，很难指定有效的协方差函数（例如，涡流动力学、密度依赖性增长等）。此外，这种基于二阶矩的方法限制了非线性和非高斯过程。实际上，如 <code>图 1</code> 所示，在预测未来的多个时间步和/或必须插补感兴趣时空域中的大片空白时，这些约束表现得最为明显。</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20221206171221-00cc.webp" alt="Figure01"></p>
<blockquote>
<p>图 1：来自 SeaWiFS 卫星的海洋彩色图像——请注意，海洋水色是海洋中浮游植物初级生产力的代表。左侧子图显示了一个示意图框，表示由于云层覆盖而经常遇到的缺失观测结果。右侧子图显示该区域存在中尺度涡流（中等空间尺度高度非线性环流特征）。这说明了尝试将传统的基于插值的空间或时间预测方法用于复杂过程存在较大的挑战。</p>
</blockquote>
<h3 id="2-1-动态时空模型（DSTM）">2.1 动态时空模型（DSTM）</h3>
<p>统计中时空过程建模的动态方法基于将当前时间的空间过程调节到最近的过去（即马尔可夫假设）的想法。该模型主要关注指定空间场随时间的演变。这种对空间过程演变的指定方式描述了环境过程的病因学。当一个人对感兴趣的过程有一些基础知识以帮助估计控制演化的迁移算子时，这种指定传统上非常有效（例如，Wikle 和 Hooten，2010）。这些模型通常在预测未来的多个时间步长和/或预测没有观测值的大空间区域时最有效。</p>
<p>一般DSTM中的 <code>数据模型</code> 可以写成</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="script">H</mi><mo stretchy="false">(</mo><msub><mi>Y</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>d</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub><mo separator="true">,</mo><msub><mi>ϵ</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>t</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>T</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(6)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">z_t(·) = \mathcal{H}(Y_t(·), \boldsymbol{\theta}_{d,t}, \epsilon_t(·)), t = 1, \ldots, T \tag{6}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathcal" style="margin-right:0.00965em;">H</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">))</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span><span class="tag"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">6</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z_t(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 对应于时间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 的数据，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_t(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 对应感兴趣的隐过程，具有线性或非线性映射函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">H</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{H}(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.00965em;">H</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span>，将数据与隐过程相关联。数据模型误差由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϵ</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon_t(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>  给出，数据模型参数由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>d</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_{d,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 表示。这些参数通常可以在空间和/或时间上发生变化。此处以及上述描述性模型中存在的一个重要假设是，当以真实过程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_t(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 和参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>d</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_{d,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 为条件时，数据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z_t(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 在时间上是独立的 （注意，按照动态模型的惯例，我们在这里将时间索引表示为下标）。</p>
<p>DSTM 最重要的组成部分是 <code>动态过程模型</code>。人们可以通过马尔可夫假设，并利用条件独立性来简化该过程（例如，以最近的过去为条件，该过程独立于更久远的过去的过程）。例如，一阶马尔可夫过程可以写成</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>Y</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="script">M</mi><mo stretchy="false">(</mo><msub><mi>Y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>p</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub><mo separator="true">,</mo><msub><mi>η</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>t</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(7)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">Y_t(·) = \mathcal{M}(Y_{t−1}(·), \boldsymbol{\theta}_{p,t}, \eta_t(·)), t = 1, 2, . . .  \tag{7}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathcal">M</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">))</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span></span><span class="tag"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">7</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">M</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal">M</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是状态转移算子（线性或非线性），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>η</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_t(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是噪声（误差）过程，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>p</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_{p,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 是可能随时间和/或空间变化的过程模型参数。请注意，这里我们假设时间是离散的且间隔相等（尽管这可以放宽）。例如，一个线性演化方程可以写成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Y</mi><mi>t</mi></msub><mo>=</mo><msub><mrow><mi mathvariant="bold">M</mi><mi mathvariant="bold">Y</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{Y}_t = \mathbf{MY}_{t−1} + \boldsymbol{\eta}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0288em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8944em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">MY</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msub><mi mathvariant="bold">C</mi><mi>η</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{\eta}_t \sim  \text{Gau}(\mathbf{0}, \mathbf{C}_{\eta})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{Y}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0288em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是一个对应于空间位置的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n × 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 向量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">M</mi></mrow><annotation encoding="application/x-tex">\mathbf{M}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">M</span></span></span></span> 是一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n × n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 维的转移矩阵，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">C</mi><mi>η</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{C}_{\eta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n × n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 新误差的协方差矩阵（本例为空间协方差矩阵）。通常，我们还会指定一个初始状态的分布 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>p</mi><mo separator="true">,</mo><mn>0</mn></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[Y_0(·)|\boldsymbol{\theta}_{p,0}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>。</p>
<p>最后，要么直接估计 <code>式 (6)</code> 和 <code>式(7)</code> 中的参数，要么为它们分配概率分布。下文将看到，DH-DSTM 框架的一个重要部分，就是将这些参数建模为过程。</p>
<h3 id="2-2-基函数表示">2.2 基函数表示</h3>
<p>时空建模的描述性方法和动态方法都存在维数灾难问题。在描述性情况下，我们需要能够有效地计算逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">C</mi><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{C}^{-1}_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>；在动态情况下，我们需要能够估计转移算子中的参数（例如，线性情况下的转移矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">M</mi></mrow><annotation encoding="application/x-tex">\mathbf{M}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">M</span></span></span></span> )。如果空间位置（数据和/或预测）的数量很大，这将具有挑战性。有许多方法可以缓解这些问题（例如，请参阅概述，Heaton 等，2018 年，主要讨论空间模型），其中一种两者都有的共同方法是采用基函数表示。</p>
<p>考虑时空过程的有限维基展开：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold">x</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal">′</mo></msup><mi mathvariant="bold-italic">β</mi><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mi>α</mi></msub></munderover><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ν</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(8)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">Y (\mathbf{s};t) = \mathbf{x}(\mathbf{s};t)^\prime \boldsymbol{\beta} + \sum^{n_\alpha}_{i=1} \alpha_i(t)\phi_i(\mathbf{s}) + \nu(\mathbf{s};t)  \tag{8}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathbf">x</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9402em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6625em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3111em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:2.9402em;vertical-align:-1.2777em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">8</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo><mo>:</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>n</mi><mi>α</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\phi_i(\mathbf{s}) : i = 1, . . . , n_{\alpha}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 为基函数, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>n</mi><mi>α</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\alpha_i(t) : i = 1, . . . , n_{\alpha}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是相应的随机展开系数，而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nu(\mathbf{s};t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 是一个相对简单的时空过程，用于表示剩余的精细尺度时空随机变化。请注意，我们可以考虑在空间和时间或仅时间中索引的基函数（例如，参见 Wikle 等人，2019）。</p>
<p>当然，在 Mercer 定理和高斯过程的 Karhunen-Loeve 分解上下文中，协方差函数、基函数和核之间存在众所周知的联系（例如，参见 Rasmussen 和 Williams，2006 年）。请注意，它们允许我们以计算有效的方式通过边缘化来构建复杂性。例如，对于线性混合模型理论，我们可以编写（以向量/矩阵形式）如下条件模型：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">Y</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">α</mi><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mi mathvariant="bold-italic">β</mi><mo>+</mo><mi><mrow><mi mathvariant="bold">Φ</mi><mi mathvariant="bold-italic">α</mi></mrow></mi><mo separator="true">,</mo><msub><mi mathvariant="bold">C</mi><mi>ν</mi></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi mathvariant="bold-italic">α</mi><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mi mathvariant="bold">0</mi><mo separator="true">,</mo><msub><mi mathvariant="bold">C</mi><mi>α</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Y}|\boldsymbol{\alpha} \sim  \text{Gau}(\mathbf{X} \boldsymbol{\beta} + \boldsymbol{\Phi \alpha}, \mathbf{C}_{\nu})\\
\boldsymbol{\alpha} \sim  \text{Gau}(\boldsymbol{0}, \mathbf{C}_{\alpha})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span><span class="mord boldsymbol">α</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
<p>然后，积分（边缘化）随机效应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">α</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span></span></span></span> 引起依赖性，可得：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">Y</mi><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mi mathvariant="bold-italic">β</mi><mo separator="true">,</mo><mi mathvariant="bold">Φ</mi><msub><mi mathvariant="bold">C</mi><mi>α</mi></msub><msup><mi mathvariant="bold">Φ</mi><mo mathvariant="normal">′</mo></msup><mo>+</mo><msub><mi mathvariant="bold">C</mi><mi>ν</mi></msub><mo stretchy="false">)</mo><mtext>。</mtext></mrow><annotation encoding="application/x-tex">\mathbf{Y} \sim  \text{Gau}(\mathbf{X} \boldsymbol{\beta}, \boldsymbol{\Phi}\mathbf{C}_{\alpha}\boldsymbol{\Phi}^\prime + \mathbf{C}_{\nu})。
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0779em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord cjk_fallback">。</span></span></span></span></span></p>
<p>即我们通过已知基函数和随机效应中的依赖关系，构造了边缘协方差矩阵：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">C</mi><mi>y</mi></msub><mo>=</mo><mi mathvariant="bold">Φ</mi><msub><mi mathvariant="bold">C</mi><mi>α</mi></msub><msup><mi mathvariant="bold">Φ</mi><mo mathvariant="normal">′</mo></msup><mo>+</mo><msub><mi mathvariant="bold">C</mi><mi>ν</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{C}_y = \boldsymbol{\Phi}\mathbf{C}_{\alpha}\boldsymbol{\Phi}^\prime + \mathbf{C}_{\nu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9722em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9779em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</p>
<p>在这种情况下，主要的时空依赖结构来自描述性方法下的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">C</mi><mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{C}_{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，或动态方法下的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo>=</mo><msub><mi mathvariant="bold">M</mi><mi>α</mi></msub><msub><mi mathvariant="bold-italic">α</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t} = \mathbf{M}_{\alpha} \boldsymbol{\alpha}_{t-1} + \boldsymbol{\eta}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8944em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathbf">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>。然后，当人们认识到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\boldsymbol{\alpha}_{t}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo separator="true">;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Y (\mathbf{s};t)\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mopen">(</span><span class="mord mathbf">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)}</span></span></span></span> 更简单时，基函数的计算优势就出现了，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">C</mi><mi>α</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{C}^{-1}_{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> 和/或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">M</mi><mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{M}_{\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 很容易获得。当使用低秩系统（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>α</mi></msub><mo>≪</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n_{\alpha} \ll n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>）或存在用于操纵基函数和/或随机效应的有效算法时（例如，参见 Cressie 和 Wikle，2011），就会发生这种情况。基函数方法对于时空建模非常有用，但仍有许多情况需要对随机效应进行更复杂的过程描述。这最好从分层建模的角度考虑</p>
<h2 id="3-深层分层统计模型">3 深层分层统计模型</h2>
<p>什么是深度模型？尽管可能没有普遍同意的答案，但一种普遍的观点认为，深度模型的结构使响应（输出）由一系列模型的链接给出：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Response (Output)</mtext><mo>←</mo><msub><mi>m</mi><mn>1</mn></msub><mo>←</mo><msub><mi>m</mi><mn>2</mn></msub><mo>←</mo><mo>…</mo><mo>←</mo><msub><mi>m</mi><mi>L</mi></msub><mo stretchy="false">(</mo><mo>←</mo><mtext>Input</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Response (Output)} \leftarrow m_1 \leftarrow m_2 \leftarrow \ldots \leftarrow  m_L(\leftarrow  \text{Input})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Response (Output)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Input</span></span><span class="mclose">)</span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mi mathvariant="normal">ℓ</mi></msub></mrow><annotation encoding="application/x-tex">m_\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对应于第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span> 个模型。在统计数据中，这可能最好由贝叶斯分层建模框架表示（例如，参见 Gelman 和 Hill，2006 年；Gelman 等人，2013 年）。在这种场景中，输入不只是可以包含在模型的最底层，而是可以在任何阶段，包括最顶部（注：传统深度模型通常采用自底向上的绘制方式）。特别是，在环境统计背景下，Berliner (1996)、Wikle 等 (1998) 以及 Cressie 和 Wikle (2011) 的分层建模范式考虑了以下的通用分布/模型：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Data Models</mtext><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">[</mo><mtext> data | process, data parameters </mtext><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Process Models</mtext><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">[</mo><mtext> process | process parameters </mtext><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Parameter Models</mtext><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">[</mo><mtext> data and process parameters </mtext><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\text{Data Models}: &amp;[\text{ data | process, data  parameters }] \\
\text{Process Models}: &amp;[\text{ process | process parameters }] \\
\text{Parameter Models}: &amp;[\text{ data  and  process parameters }]
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5em;vertical-align:-2em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Data Models</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Process Models</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Parameter Models</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">[</span><span class="mord text"><span class="mord"> data | process, data parameters </span></span><span class="mclose">]</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">[</span><span class="mord text"><span class="mord"> process | process parameters </span></span><span class="mclose">]</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">[</span><span class="mord text"><span class="mord"> data and process parameters </span></span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>对于推断和预测，我们需要评估后验分布：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Posterior</mtext><mo>:</mo><mo stretchy="false">[</mo><mtext> process, parameters | data </mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{Posterior}: [ \text{ process, parameters | data }]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Posterior</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord text"><span class="mord"> process, parameters | data </span></span><span class="mclose">]</span></span></span></span></span></p>
<p>根据贝叶斯定理，后验分布与上面给出的数据、过程和参数的联合分布（根据条件概率定义，就是三者的乘积）成正比。通常每个层级都可以有多个子阶段，进而增加了模型深度。Berliner (1996) 分层模型范式的关键在于： <strong>尽可能避免对二阶结构建模</strong>。也就是说，应当将建模的重点放在条件均值上，通过边缘化建立依赖性（复杂性）。因此，这些链接的条件模型通常自上而下，而输入则更接近顶层（数据）级别，尽管理论上它们可以出现在任何层级的输入中。下一节以一个用于复杂时空建模的通用 DH-DSTM 深度模型为例进行说明。</p>
<h3 id="3-1-深度分层动态时空模型（DH-DSTMs）">3.1 深度分层动态时空模型（DH-DSTMs）</h3>
<p>在这里，我们概述了一个原型 DH-DSTM。为简单起见，并与 <code>第 4 节</code> 中的深度机器学习模型进行比较，该模型是在离散时间和空间的背景下呈现的，尽管时间和/或空间可以更普遍地被认为是连续的。对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn><mtext>，</mtext><mo>…</mo><mo separator="true">,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t = 1，\ldots,T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord cjk_fallback">，</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span></p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Data Model</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold">z</mi><mi>t</mi></msub><mi mathvariant="normal">∣</mi><msub><mi mathvariant="bold">Y</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>h</mi></msub><mo>∼</mo><mi mathvariant="script">D</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">H</mi><mi>t</mi></msub><msub><mi mathvariant="bold">Y</mi><mi>t</mi></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Conditional Mean</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">Y</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="bold-italic">μ</mi><mi>t</mi></msub><mo>+</mo><mi mathvariant="bold">Φ</mi><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo>+</mo><msub><mi mathvariant="bold-italic">ν</mi><mi>t</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Process Mean</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold-italic">μ</mi><mi>t</mi></msub><mo>=</mo><msub><mi mathvariant="bold">W</mi><mi>t</mi></msub><msub><mi mathvariant="bold-italic">θ</mi><mi>μ</mi></msub><mo>+</mo><msub><mi mathvariant="bold-italic">γ</mi><mi>t</mi></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Dynamic Process</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">α</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">x</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>α</mi></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>“Residual” Process</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">ν</mi><mi>t</mi></msub><mi mathvariant="normal">∣</mi><msub><mi mathvariant="bold-italic">θ</mi><mi>ν</mi></msub><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Regularization Priors</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>α</mi></msub><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">ζ</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Parameters</mtext><mo>:</mo><mspace width="1em"/></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">[</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>h</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>ν</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>μ</mi></msub><mo separator="true">,</mo><mi mathvariant="bold-italic">ζ</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\text{Data Model}:\quad &amp;\mathbf{z}_t | \mathbf{Y}_t, \boldsymbol{\theta}_h \sim  \mathcal{D}(\mathbf{H}_t\mathbf{Y}_t; \boldsymbol{\theta}_h) \tag{9}\\ 
\text{Conditional Mean}: \quad &amp;f (\mathbf{Y}_t) = \boldsymbol{\mu}_t + \boldsymbol{\Phi} \boldsymbol{\alpha}_t + \boldsymbol{\nu}_t \tag{10}\\
\text{Process Mean}: \quad &amp;\boldsymbol{\mu}_t = \mathbf{W}_t\boldsymbol{\theta}_{\mu} + \boldsymbol{\gamma}_t \tag{11}\\
\text{Dynamic Process}: \quad &amp;\boldsymbol{\alpha}_{t} = g(\boldsymbol{\alpha}_{t−τ} , \mathbf{x}_{t−τ} ; \boldsymbol{\theta}_\alpha; \boldsymbol{\eta}_t) \tag{12}\\
\text{“Residual” Process}: \quad &amp;[\boldsymbol{\nu}_t|\boldsymbol{\theta}_{\nu}]\tag{13}\\ 
\text{Regularization Priors}: \quad &amp;[\boldsymbol{\theta}_\alpha|\boldsymbol{\zeta}]\\
\text{Parameters}: \quad &amp;[\boldsymbol{\theta}_h, \boldsymbol{\theta}_{\nu}, \boldsymbol{\theta}_{\mu}, \boldsymbol{\zeta}] 
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.5em;vertical-align:-5em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Data Model</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Conditional Mean</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Process Mean</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Dynamic Process</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">“Residual” Process</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Regularization Priors</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:1.34em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Parameters</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0288em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0288em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0288em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06898em;">ν</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06389em;">γ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06898em;">ν</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06215em;">ζ</span></span></span><span class="mclose">]</span></span></span><span style="top:1.34em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06215em;">ζ</span></span></span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">9</span></span><span class="mord">)</span></span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">10</span></span><span class="mord">)</span></span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">11</span></span><span class="mord">)</span></span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">12</span></span><span class="mord">)</span></span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">13</span></span><span class="mord">)</span></span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span></span></span><span style="top:1.34em;"><span class="pstrut" style="height:3em;"></span><span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5em;"><span></span></span></span></span></span></span></span></span></p>
<p><code>式 (9)</code> 的数据模型指定了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">z</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{z}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的分布，它是时间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 的空间参考数据向量。具体来说，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">D</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 指一些通用分布（例如，指数族，需要特定于问题），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">H</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是将隐过程位置映射到数据位置的映射矩阵，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">Y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{Y}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.02875em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0288em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 时刻的隐过程向量， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mi>h</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是数据模型参数。该数据模型中的一个重要假设是：以隐时空过程作为条件的观测相互独立，并且观测误差的结构相对简单（即观测误差也被视为相互独立）；因为大部分依赖性都应当归因于隐时空过程。另请注意，可以像一般的 Berliner (1996) 框架一样轻松地容纳多个数据（输入）源。</p>
<p><code>式 (10)</code> 的条件均值指定变换（链接函数）<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f (·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">μ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\mu}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 是随时间变化的空间 “趋势”（注意，这可能取决于输入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{x}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ）；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Φ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\Phi}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span></span></span></span> 是空间基函数矩阵（提供降维）；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是一个潜在的动态随机过程（ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>α</mi></msub><mo>≪</mo><msub><mi>n</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">n_{\alpha} \ll n_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> ），而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ν</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\nu}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06898em;">ν</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是一个非动态时空随机过程（如下所述）。模型这部分最重要的假设是潜在动力过程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\boldsymbol{\alpha}_{t}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是低维的。</p>
<p>过程均值在<code>式 (11)</code> 中给出，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">W</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 包含适应趋势、偏差、季节性等的协变量输入，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mi>μ</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_{\mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 是相关参数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">γ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\gamma}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06389em;">γ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 是误差过程（通常为高斯分布）。请注意，如有必要，可以在此处考虑更灵活的协变量函数（即如在广义加性模型中），但数据中的大部分复杂结构是由于下面描述的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 项。另请注意，假定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">γ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\gamma}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06389em;">γ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 的均值为零，并且通常假定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">γ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\gamma}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06389em;">γ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 在时间和空间上是独立的。</p>
<p>模型的动态部分由 <code>式(12)</code> 给出，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是转移算子（在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t−τ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6528em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 和输入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{x}_{t−τ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6528em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 中可能是非线性的），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是参数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\eta}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 是噪声过程（通常假设为是高斯分布且均值为零，依赖结构取决于具体问题）。该模型可以说是 DH-DSTM 中最重要的部分。它通常是高度参数化的，并且如果信息可用，可以根据机制模型来制定，或者至少由此类模型驱动。无论如何，至关重要的是，该动力学模型允许 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>  的元素随时间进行交互（参见 Wikle 等人，2019 年，第 5 章中的讨论）。例如，考虑 Wikle 和 Hooten (2010) 的一般二次非线性 (GQN) 模型：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>α</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><msubsup><mi>θ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow><mi>L</mi></msubsup><msub><mi>α</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><munderover><mo>∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msubsup><mi>θ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>k</mi><mi mathvariant="normal">ℓ</mi></mrow><mi>Q</mi></msubsup><msub><mi>α</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>α</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">ℓ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>g</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>η</mi><mi>t</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(14)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">{\alpha}_{t}(i) = \sum^p_{j=1} \theta^L_{i,j} \alpha_{t−τ} (j) +  \sum^p_{k=1}  \sum^k_{\ell=1} \theta^Q_{i,k \ell} {\alpha}_{t−τ} (k) g({\alpha}_{t−τ} ( \ell) , \mathbf{x}_t; \boldsymbol{\theta}_g) + \eta_t(i) \tag{14}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1123em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6985em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.1382em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6985em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9592em;"><span style="top:-2.3987em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4374em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">ℓ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:3.2499em;vertical-align:-1.4138em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">14</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中单个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 分量的演变由线性相互作用（右侧第一项，参数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>θ</mi><mi>L</mi></msup></mrow><annotation encoding="application/x-tex">θ^L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span></span></span></span></span></span></span>）和二次交互作用（右侧第二项，参数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>θ</mi><mi>Q</mi></msup></mrow><annotation encoding="application/x-tex">θ^Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">Q</span></span></span></span></span></span></span></span></span></span></span>）和噪声项控制。函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo separator="true">;</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·;·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅;⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是一个转移函数，用于限制由非线性交互作用引起的爆炸式增长。该模型由物理和生物科学中的各种过程驱动（参见 Wikle 和 Hooten，2010 年），并且非常灵活。不过，此模型被 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(p^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 个参数严重过度参数化了，需要基于科学的硬阈值或正则化/稀疏性才能实际运行。</p>
<p>残差时空过程如式(13)所示，其分布由具体问题决定。例如，一个有用的参数化是假设另一个基展开形式，例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ν</mi><mi>t</mi></msub><mo>=</mo><msub><mi><mrow><mi mathvariant="bold">Ψ</mi><mi mathvariant="bold-italic">ω</mi></mrow></mi><mi>t</mi></msub><mo>+</mo><msub><mi mathvariant="bold-italic">ξ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\nu}_t = \boldsymbol{\Psi\omega}_t + \boldsymbol{ξ}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06898em;">ν</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">Ψ</span><span class="mord boldsymbol" style="margin-right:0.03704em;">ω</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03021em;">ξ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Ψ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\Psi}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Ψ</span></span></span></span></span></span> 是空间基函数矩阵，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ω</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\omega}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">ω</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是展开系数，而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ξ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{ξ}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03021em;">ξ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 是一个简单的误差过程（例如 Wikle 等人，2001 年）。这里的假设是复杂的时空动力学被 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 捕获，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ω</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\omega}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">ω</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 将有一个简单的分布（例如，可能具有简单时间依赖性但在“<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ω</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\omega}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">ω</span></span></span></span></span></span> 空间”中独立的高斯分布），并且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ξ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{ξ}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03021em;">ξ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span> 将在时间和空间上独立。</p>
<p>如上所述，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\alpha}_{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的动态模型可能过度参数化并且通常需要正则化。此处可以使用贝叶斯模型上下文中任何常用的正则化方法（例如，随机搜索变量选择、spike-and-slab、马蹄铁先验等；Fan 和 Lv（例如，参见 2010））。最后，我们需要其余参数的分布或固定值。重要的是，在深度 DH-DSTM 中，这些参数本身可能是“过程”（空间或时间），并且可能包括对各种外生输入变量的依赖。这种深度/复杂贝叶斯模型的实现通常是通过特定于问题的 MCMC 算法，尽管最近有人尝试在变分贝叶斯上下文中考虑相当复杂的 DSTM（例如，Quiroz 等人，2018 年）。一般而言，MCMC 实施可能非常耗时，并且需要大量数据、先验信息和计算资源才能成功。</p>
<h3 id="3-2-DH-DTSM-示例：海洋水色">3.2 DH-DTSM 示例：海洋水色</h3>
<p>Leeds等 (2014) 使用 DH-DSTM 模型执行时空预测以填补 SeaWiFS 海洋水色观测中的空白，类似于 <code>图 1</code> 中所示的问题。他们考虑了一个多元模型，除了 SeaWiFS 观测之外，还包括海面高度(SSH) 和海面温度 (SST) 从区域海洋模型系统 (ROMS) 输出，该系统与低营养生态系统的生物地球化学模型相结合。他们实现了一个类似于 <code>式 (14)</code> 的降维 GQN 过程模型作为 ROMS 模型的仿真器（例如，ROMS 模型输出用于训练 GQN 模型的先验分布——类似于下面描述的机器学习预训练）。详细信息可以在 Leeds 等 (2014) 中找到。如 <code>图 2</code> 所示，该模型能够预测浮游植物场中的涡流，尽管事实上阿拉斯加湾沿岸地区的云层覆盖在 SeaWiFS 数据中留下了持续的空白。</p>
<p>重要的是，模型的概率性质产生了不确定性度量，表明最大的不确定性不是该区域存在涡流，而是它的精确位置。</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20221206171135-2e54.webp" alt="Figure02"></p>
<blockquote>
<p>图 2：三个八天时间段的对数转换 SeaWiFS 海洋水色观测图（顶行）、DH-DSTM 后验平均值（第二行）和 DH-DSTM 后验标准差（第三行）：6 月 2 日， 2002年至2002年6月9日（左栏），2002年6月10日至2002年6月17日（中栏），2002年6月18日至2002年6月25日（右栏）</p>
</blockquote>
<h2 id="4-深度神经网络模型">4 深度神经网络模型</h2>
<p>深度神经模型的开发和应用在过去十年中发展迅速。在 Goodfellow 等人的教科书中可以找到广泛的概述。 (2016) 和 Aggarwal (2018)。本节的目的不是提供如此全面的处理，而是简要概述以方便与 DSTM 的连接。我们描述了简单的前馈神经网络 (NN)、深度前馈神经网络 (DNN)、卷积神经网络 (CNN) 和递归神经网络 (RNN)。这为讨论时空数据的深度机器学习模型提供了背景，我们称之为深度神经 DSTM (DN-DSTM)。</p>
<h3 id="4-1-神经网络">4.1 神经网络</h3>
<p>略。</p>
<!-- 我们从一个非常简单的神经网络开始，称为单隐藏层前馈网络或单层感知器。假设我们有一个 p 维输入向量 x 和一个 m 维响应（输出）向量 z（但请注意，在大多数非线性回归和二元分类问题中 m = 1）。我们现在寻求一个非线性模型来响应给定的输入通过“隐藏层”给出的转换

yj = g( p \sum i=0 wjixi), j = 1, . . . , , ,

其中 yj 是隐藏变量，{wji} 是权重（参数），其中 wj0 是偏差或偏移（截距）参数（注意，x0 ≡ 1），g(·) 是激活函数（例如，双曲线正切、径向基函数、修正线性单元、softmax 等）。 “输出层”然后由下式给出：

k = g_o( J \sum j=0 vkjyj), k = 1, . . . , 米

其中g_o(·)是激活函数（可以是恒等函数），y0 ≡ 1，{vkj}是输出权重，包括偏移量。可以将隐藏层变换视为输入的基础扩展，在这种情况下，我们可以简单地将模型写为：

zk(x; W, V) = go   J \sum j=0 vkj g( p \sum i=0 wjixi)  

其中\mathbf{W}= {wji}，V = {vkj}，我们注意到该模型中没有明确的误差项。

与传统的非线性回归一样，为了估计此类模型中的参数（即“训练网络”），我们根据 {W, V}（例如，平方误差、交叉熵）选择目标函数，然后通常使用梯度基于的方法来获得\mathbf{W}和 V 参数的参数估计。传统上，神经网络社区使用反向传播来做到这一点。反向传播基于应用链式法则计算梯度，由于模型的层次/组合性质，这种方法简单明了且有用。这是在具有局部性重要特征的两次通过算法中实现的，因为每个隐藏单元仅向共享连接的单元传递信息和从共享连接的单元接收信息。这有助于并行计算环境中的计算，这对于大型数据集很重要。

因为目标函数由训练数据的总和组成，可能非常大，所以梯度的计算可能很昂贵。此外，训练样本中可能存在冗余数据。缓解这些问题的一种方法是考虑最小化预期损失，这可以很容易地通过训练样本的小随机样本（即小批量）的平均值来估计。这是随机梯度下降 (SGD) 的本质，它是现代神经计算中的主导范式（例如，参见 Goodfellow 等人，2016 年；Aggarwal，2018 年）。它不仅对大数据有帮助，而且 SGD 还有助于保持优化不受局部最小值的影响。即使是这些简单的一层 NN 也容易过度拟合，因此它们包含某种形式的正则化很重要。例如，可以将权重的 L2（ridge）惩罚添加到目标函数（称为“权重衰减”）或可以添加 L1（套索）惩罚，这称为“权重消除”。
-->
<h3 id="4-2-深度前馈网络-DNN">4.2 深度前馈网络 (DNN)</h3>
<p>略。</p>
<!-- 许多涉及大数据的问题，如声学处理、图像处理、自然语言处理等，结构非常复杂，为新一代深度学习算法的发展提供了动力。这些通常是具有许多隐藏层的神经网络，一层的输出成为下一层的输入。我们将每层中的单元数视为网络的宽度，将层数视为网络的深度。兼具宽度和深度提供了一个非常灵活的学习环境，但也带来了许多挑战。 DNN 利用了许多技术创新，这些创新是深度学习在大型数据集中的许多当前应用的基础（例如，Hinton 等人，2012 年）。可以在 Goodfellow 等人中找到全面的概述。 (2016) 和 Aggarwal (2018)。

一个基本的 DNN 可以表示为：

z(x) = go,VL(gWL(···gW1(x)))

其中 go,VL 是权重为 VL 的输出函数，gW \ell 是依赖于参数\mathbf{W}\ell 的非线性激活函数，如 (15) 中所示。 DNN 的分层性质在一个简单的示例中显而易见，该示例具有两个隐藏层和一个输出层（具有恒等输出函数）：

z = Vy2, y2 = g(W2y1 + w0,2), y1 = g(W1x + w0,1),

其中隐藏向量 y1 和 y2 的维度可能不同。以类似于单隐藏层模型的方式进行反向传播训练。一个重大的挑战出现了，因为随着深度的增加，这个模型中通常有大量的参数，这使得 DNN 难以训练。特别是，在标记响应数量相对较少的传统应用中，存在几个问题：例如，（1）对隐藏层数和隐藏单元数的敏感性； (2) 对其他调整参数的敏感性（如果可行，可以使用交叉验证）； (3) 对权重初始值极度敏感； (4) 在标准计算平台上优化非常缓慢； (5) 拟合模型有过度拟合的倾向。

对基本的基于梯度的优化的修改使这些模型能够适应大型数据集。最早的“突破”之一是生成式预训练。本质上，这是在执行反向传播优化之前尝试“大致”获取参数。生成预训练背后的关键思想是一次学习一层，隐藏单元在一个级别预测，然后作为训练下一个级别的输入。这是生成性的，因为它从底部开始，一次构建一层——最终产生响应。这里重要的是相关的权重估计（近似值）只是作为反向传播算法的起始值。反向传播算法然后使用所有信息并微调估计值。重要的是要注意生成预训练不使用标记的响应，因此它是无监督的。这为参数提供了更大的自由度并防止过度拟合，但反向传播算法使用标记的响应来获得最终估计。主要的生成模型是受限玻尔兹曼机 (RBM) 和自动编码器（例如 Goodfellow 等人，2016 年）。这两种方法的优点是它们具有无向连接（归纳权重趋向于改善泛化的最小值，例如，参见 Erhan et al. (2010)），很容易堆叠（因此一个的输出可以形成输入另一个），并且不受监督。

除了生成式预训练之外，其他因素已被证明对前馈 DNN 的实施很重要，例如：（1）使用未标记的数据来训练模型（这允许更大的灵活性）； (2) 使用 node dropout 进行正则化（收缩），这对过度拟合有很大帮助（本质上，每个节点在被训练时都有可能出现在模型中）； (3) 高效计算（即，这些模型需要大量的计算能力来拟合——分布式和并行计算是必不可少的，近年来基于图形处理单元（GPU）的并行计算使这成为可能）； (4) 修正线性单元 (ReLU) 激活函数，ReLU (x) = max(0, x)，可以加快训练速度。当这些模型还可以利用时间和空间固有的多尺度特性时，它们可能已经显示出最大的成功，就像 CNN 和 RNN 一样。 -->
<h3 id="4-3-卷积神经网络（CNN）">4.3 卷积神经网络（CNN）</h3>
<p>略。</p>
<!-- CNN 是深度学习领域最成功的案例之一，尤其是在图像处理方面。回想一下二维离散卷积的定义：

k[x, y] ∗ z[x, y] = ∞ \sum i=−∞ ∞ \sum j=−∞ k[i, j]gzx − i, y − j],

实际上，由于图像中的像素数量有限，因此总和是有限的。我们可以将 k[ ] 视为应用于空间图像 z[ ] 元素的核权重函数。根据内核权重，在进行卷积后可以获得与图像相关的不同属性（参见图 3）。请注意，在实践中，彩色图像的像素由红色、绿色和蓝色 (RGB) 像素的组合表示，因此最好将图像视为张量。可以很容易地修改卷积函数以对张量值像素进行操作。

CNN 认为图像的卷积具有已知的权重；为每个级别完成多次以获得不同的“特征图”。也就是说，CNN 不是指定核函数，而是以每个卷积都有一组核权重的方式来学习它们，因此权重在图像中共享（这导致必须的参数数量显着减少）学习）。这个卷积步骤之后是一个池化层（或者，子采样或下采样）。池化层考虑来自卷积步骤的一个小矩形块，并以某种方式对其进行子采样或聚合以产生单个输出。也许最常见的池化只是取块最大值（称为“最大池化”）。池化是有益的，因为它有助于降低 CNN 对输入翻译的敏感度。重要的是，它还减小了下一级图像的大小。图 3 中的右侧子图说明了池化。

CNN 的一般结构具有交替的卷积层和池化层，最后一层是完全连接的（如在 DNN 中）。通常，（1）在卷积阶段通过多个内核权重矩阵创建多个特征图； (2) 卷积图像进入非线性激活函数——通常是 ReLU 函数； (3) 池化可以发生在多个特征图上。 CNN 需要估计的关键阶段是卷积步骤。让 y \ell−1 i,j 对应于卷积步骤的输入。然后卷积由下式给出：

y \ell i,j = gp(g( \sum a \sum b k( \ell) a,by \ell−1 i+a,j+b)),

其中 gp(·) 是池化函数，g(·) 是非线性激活（例如 ReLU），kab 是必须学习（估计）的内核权重。请注意，池化层很简单，不需要学习。与 DNN 一样，模型的其他组件的训练是通过梯度下降反向传播算法完成的，具有与第 4.2 节中描述的相同的增强功能。
-->
<h3 id="4-4-循环神经网络（RNNs）">4.4 循环神经网络（RNNs）</h3>
<p>略。</p>
<!-- 循环神经网络 (RNN) 最初是在 1980 年代开发的，用于处理序列数据。近年来，它们已经发展成为最常用和最成功的深度学习方法之一，特别是对于语言处理应用（例如，语音识别、文本生成、机器翻译等）。这些模型类似于动力系统的多变量状态空间模型，正如人们可能在时间序列、计量经济学或时空统计中看到的那样。考虑一个经典动力系统：yt = \mathcal{M}(yt−1; θ)，其中 yt 表示系统在时间 t 的状态。这被认为是“循环的”，因为时间 t 的状态指回时间 t − 1 的状态，等等。我们可以将其重写为所谓的“展开”形式，yt = \mathcal{M}(\mathcal{M}(\mathcal{M}(yt−3 ;θ);θ);θ···)。请注意，参数 θ 在所有状态之间共享。然后将隐藏状态与观测方程中的输出 z_t 相关联。与状态空间模型一样，RNN 设置中的状态可能取决于外部输入 xt。

因此，最基本的（“vanilla”）RNN 由下式给出

z_t = g_o(Vyt) yt = g(Wyt−1 + Uxt),

其中 g_o(·) 是输出函数，g(·) 通常是双曲正切激活函数，U、V 和\mathbf{W}是权重矩阵（通常也包含偏差/偏移项）。与其他 NN 一样，为了估计参数，我们定义了一个损失函数，并希望通过 SGD 反向传播进行优化。然而，RNN 的情况很复杂，因为参数在时间上是通用的，因此必须实施时间反向传播 (BPTT) 算法（例如，参见 Aggarwal 的概述，2018 年）。为这种普通 RNN 实施 BPTT 优化的一个严峻挑战是所谓的消失梯度/爆炸梯度问题。也就是说，随着每个时间步的移动，梯度会变得越来越小（通常）或越来越大（在 RNN 实现中通常有很多时间步）。

对 RNN 进行了许多修改，这些修改已被指定用于缓解消失/爆炸梯度问题。也许最常见的方法包括打破时间结构的门，允许在特定时间步长考虑过去的一些隐藏状态，而其他隐藏状态则被遗忘。例如，长短期记忆 (LSTM) RNN 使用门来创建具有不会消失或爆炸的梯度的时间路径（Hochreiter 和 Schmidhuber，1997）。基本的 LSTM 结构如下（注意，ot  是 Hadamard（逐元素）积）：

Output: z_t = g_o(Vyt)  
Hidden State: yt = tanh(ct) ot  o Internal Memory: ct = ct−1 ot  f + g ot  i Candidate Hidden State: g = tanh(Ugxt + Wgyt−1) Output Gate: o = g(Uoxt + Woyt−1) Forget Gate: f = g(Uf xt + Wf yt−1) Input Gate: i = g(Uixt + Wiyt−1),

其中通常 g(·) 是一个 sigmoid 函数。输入门选择在时间 t 获得输入的隐藏单元，遗忘门选择之前时间的隐藏状态在时间 t 重置为 0，输出门选择将与响应相关的状态。记忆单元是至关重要的，因为它们指示何时记住或忘记以前的隐藏状态——这种记忆功能不仅有助于缓解消失/爆炸梯度问题，而且对于（遥远的）过去的事件可以影响的许多过程也是现实的。存在与干预国无关。门控循环单元 (GRU) RNN 是门控循环单元 (GRU) RNN（Cho 等人，2014 年）。

在一般实践中，门控 RNN 可能是计算密集型的，并且通常需要并行实现，并且像标准的 DNN 和 CNN 一样，需要大量的训练数据。在文献中，鉴于门控算法的复杂性，门控算法更多地被视为“黑匣子”，这有利于使它们模块化和可连接（参见第 4.5 节）。
 -->
<h3 id="4-5-回波状态网络（ESN）">4.5 回波状态网络（ESN）</h3>
<p><code>回波状态网络 (Echo State Network, ESN)</code> 是一种易于估计且通常需要较少计算资源和训练数据的 RNN （Lukosevicius 和 Jaeger，2009）：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi mathvariant="bold">z</mi><mi>t</mi></msub><mo>=</mo><msub><mi>g</mi><mi>o</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold">V</mi><mi mathvariant="bold">y</mi></mrow><mi>t</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi mathvariant="bold">y</mi><mi>t</mi></msub><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><msup><mi mathvariant="bold">W</mi><mo>∗</mo></msup><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold">U</mi><mi mathvariant="bold">x</mi></mrow><mi>t</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\mathbf{z}_t = g_o(\mathbf{Vy}_t)\\
\mathbf{y}_t = g(\mathbf{W}^*\mathbf{y}_{t−1} + \mathbf{Ux}_t)
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">Vy</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Ux</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>这看起来像上面给出的基本 RNN，但值得注意的是，权重矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">U</mi></mrow><annotation encoding="application/x-tex">\mathbf{U}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">U</span></span></span></span> 是稀疏的并且在 ESN 中是随机选择的，因此只学习了输出矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>（通过正则化）。这种在非线性变换中使用随机参数通常被称为“储层计算”。一个复杂的问题是，这种方法需要修改权重 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span></span></span></span>（此处由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>W</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 给出——见下文）以确保 “回波状态特性”，这实质上表明初始条件的影响随时间渐近减小。总体而言，ESN 极大地减少了待估计的参数并极大地简化了模型，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{y}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 只是输入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{x}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 基于随机权重的一系列随机变换，而输出函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>o</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_o(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">V</mi></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">V</span></span></span></span> 参数可以像在基本统计模型（例如，回归、逻辑、softmax）中一样进行训练。 ESN 通常需要比传统 RNN 更多的隐藏单元（即更宽）来补偿未学习权重，因此在估计 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">V</mi></mrow><annotation encoding="application/x-tex">\mathbf{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">V</span></span></span></span> 时必须应用正则化。我们将在下面 <code>第 5 节</code> 的 DSTM 上下文中更详细地讨论 ESN 模型。</p>
<h3 id="4-6-深度神经-DSTM（DN-DSTM）">4.6 深度神经 DSTM（DN-DSTM）</h3>
<p>尽管 DNN 可以与时空数据一起使用（Polson 和 Sokolov，2017 年），但它们并不总是合适的，因为它们不能自然地适应时间和空间中出现的依赖结构。然而，鉴于 CNN 和 RNN 的模块化（即它们很容易“堆叠”以形成更深层次的模型），它们可以很容易地以不同方式组合以产生时空数据的深度混合模型也就不足为奇了，如视频图像处理和图像标引（例如，Keren 和 Schuller，2016 年；Tong 和 Tanaka，2018 年）。例如，可以通过 CNN 缩小视频中的图像以找到空间特征，然后使用 RNN（通常是 LSTM）对这些特征的时间演化进行建模。在某些情况下，该框架还可用于将图像与标题（或描述）相关联（Donahue 等人，2015 年），将 CNN 用于对图像进行编码，而将 RNN 用于描述图像的单词序列进行解码。前者显然是一个时空问题，而后者显然在输出（单词序列）上是一个具有顺序结构的 “时间” 问题。一般来说，软件包模块化各种机器学习组件（例如 CNN 和 RNN）的能力允许开发人员以不同的方式组合这些层。在这里，我们感兴趣的是随时间演变的空间过程（类似于第一种情况）。这种方法已在环境科学中用于生成短时降水预报（Xingjian 等人，2015 年）。</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20221206172024-1271.webp" alt="Figure04"></p>
<blockquote>
<p>图 4：一般深度神经动态时空模型 (DN-DSTM) 的示意图。</p>
</blockquote>
<p>混合深度神经网络和动态时空模型的一般方法是使用堆叠的 RNN，但中间层会降低维度。<code>图 4</code> 中有示意性的显示，并且可以一般地写成：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">O</mi><mi mathvariant="bold">u</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">p</mi><mi mathvariant="bold">u</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">e</mi></mrow><mo>:</mo><msub><mi mathvariant="bold">z</mi><mi>t</mi></msub><mo>=</mo><msub><mi>g</mi><mi>o</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mi>L</mi></mrow></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>z</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">H</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">g</mi><mi mathvariant="bold">e</mi><mn mathvariant="bold">1</mn></mrow><mo>:</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>h</mi><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">R</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">u</mi><mi mathvariant="bold">c</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">o</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">g</mi><mi mathvariant="bold">e</mi><mn mathvariant="bold">1</mn></mrow><mo>:</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo>≡</mo><mi mathvariant="script">Q</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>r</mi><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">H</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">g</mi><mi mathvariant="bold">e</mi><mn mathvariant="bold">2</mn></mrow><mo>:</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mn>3</mn></mrow></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>h</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">R</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">u</mi><mi mathvariant="bold">c</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">o</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">g</mi><mi mathvariant="bold">e</mi><mn mathvariant="bold">2</mn></mrow><mo>:</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mi>t</mi></msub><mo separator="true">,</mo><mn>3</mn><mo>≡</mo><mi mathvariant="script">Q</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>3</mn></mrow></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>r</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi><mo separator="true">,</mo><mspace width="1em"/><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">H</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">g</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">L</mi></mrow><mo>:</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mi>L</mi></mrow></msub><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>L</mi></mrow></msub><mo separator="true">,</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">x</mi><mo>~</mo></mover><mi>t</mi></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mrow><mi>h</mi><mi>L</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mi mathvariant="bold">I</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">p</mi><mi mathvariant="bold">u</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">g</mi><mi mathvariant="bold">e</mi></mrow><mo>:</mo><mtext> </mtext><msub><mover accent="true"><mi mathvariant="bold">x</mi><mo>~</mo></mover><mi>t</mi></msub><mo>=</mo><msub><mi>g</mi><mi>I</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub><mo separator="true">;</mo><msub><mi mathvariant="bold-italic">θ</mi><mi>I</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
&amp;\mathbf{Output State}: \mathbf{z}_t = g_o ( \mathbf{y}_{t,1}, \tilde{\mathbf{y}}_{t,2}, \ldots , \tilde{\mathbf{y}}_{t,L}; \boldsymbol{\theta}_z ) \\
&amp;\mathbf{Hidden Stage 1}: \mathbf{y}_{t,1} = g ( \mathbf{y}_{t−1,1}, \tilde{\mathbf{y}}_{t,2}; \boldsymbol{\theta}_{h1} ) \\
&amp;\mathbf{Reduction Stage 1}: \tilde{\mathbf{y}}_{t,2} ≡ \mathcal{Q}(\mathbf{y}_{t,2}; \boldsymbol{\theta}_{r1}) \\
&amp;\mathbf{Hidden Stage 2}: \mathbf{y}_{t,2} = g ( \mathbf{y}_{t−1,2}, \tilde{\mathbf{y}}_{t,3}; \boldsymbol{\theta}_{h2} ) \\
&amp;\mathbf{Reduction Stage 2}: \tilde{\mathbf{y}}_t,3 ≡ \mathcal{Q}(\mathbf{y}_{t,3}; \boldsymbol{\theta}_{r2}) \\
&amp;\vdots,\quad \vdots \\
&amp;\mathbf{Hidden Stage L}: \mathbf{y}_{t,L} = g ( \mathbf{y}_{t−1,L}, \tilde{\mathbf{x}}_t; \boldsymbol{\theta}_{hL} ) \\
&amp;\mathbf{Input Stage}: \tilde{\mathbf{x}}_t = g_I ( \mathbf{x}_t; \boldsymbol{\theta}_I ) 
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.66em;vertical-align:-6.08em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.58em;"><span style="top:-9.24em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-7.74em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-6.24em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-4.74em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-3.24em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.08em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:0.42em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:1.92em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.08em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.58em;"><span style="top:-9.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">OutputState</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-7.9275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">HiddenStage1</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">h</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-6.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">ReductionStage1</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathcal">Q</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-4.9275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">HiddenStage2</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">h</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.4275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">ReductionStage2</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathcal">Q</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-1.2675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:0.2325em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">HiddenStageL</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">x</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">h</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:1.7325em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">InputStage</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">x</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:6.08em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>o</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_o(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是一个输出函数（例如，用于回归的恒等式，用于分类的 softmax 等），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>I</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_I(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是一个输入函数，可能会增加和/或转换输入向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{x}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是一些RNN 结构的类型（例如，LSTM、GRU、ESN），而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Q}(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal">Q</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是降维函数，例如 CNN 或更简单的东西，例如主成分分解或其他一些随机降维方法（例如，随机投影，宾厄姆和曼尼拉 (2001)）。每个函数中的潜在参数（权重）由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">θ</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\theta}_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">θ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 给出。在这个框架中，除了第 1 阶段的非缩减隐藏单元之外，每个缩减阶段的组件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\tilde{\mathbf{y}}_{t, \ell}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9674em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 都会影响输出。还可以让来自更深隐藏阶段的非缩减隐藏单元产生影响也直接输出，但这增加了必须学习的参数数量，通常是不必要的。最后，请注意，此模型可以更简洁地编写为输入的伸缩函数转换：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>z</mi><mi>t</mi></msub><mo>=</mo><msub><mi>g</mi><mi>o</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="script">Q</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mi mathvariant="script">Q</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>I</mi></msub><mo stretchy="false">(</mo><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">;</mo><mi mathvariant="bold">Θ</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(17)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">z_t = g_o(g(\mathcal{Q}(g(···\mathcal{Q}(g(g_I(\mathbf{x}_t)))))); \boldsymbol{\Theta}) \tag{17}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathcal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅⋅⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))))))</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Θ</span></span></span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">17</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Θ</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\Theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Θ</span></span></span></span></span></span> 表示函数中的所有各种参数（权重）。</p>
<p>这种方法的优点是它自然地适应了多个空间和时间尺度的可变性。注意，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>I</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_I(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 充当转换输入的编码器。例如，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>I</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_I(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 可能是 CNN，也可能是其他类型的降维程序（例如，主成分、拉普拉斯特征图、核卷积等）。然后 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span> 函数提取隐藏单元中的重要相关特征（根据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>I</mi></msub><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_I(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span> 的选择，这些特征可能在空间上被引用。然后，不同的 RNN 级别用于查找时间依赖性，通常在不同的尺度上及时（例如，Graves 等人，2013 年；Hermans 和 Schrauwen，2013 年）。请注意，可以省略各种级别；例如，我们可能会省略 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span> 阶段并形成一个没有中间减少阶段的堆叠 RNN（并且，反之亦然）。通常，此类模型将通过反向传播和 SGD 实现，具体取决于不同模型阶段的选择。</p>
<h3 id="4-7-DH-DSTMs和DN-DSTMs之间的连接">4.7 DH-DSTMs和DN-DSTMs之间的连接</h3>
<p>自然的问题是 <code>第 3.1 节</code> 中介绍的 DH-DSTM 与<code>第 4.5 节</code>中介绍的 DN-DSTM 相比如何？这两种范式确实有很多共同点，因为它们都试图在复杂时空依赖性建模的背景下做同样的事情。</p>
<p>也就是说，两者都在处理这样一个事实，即存在多个相互作用以描述过程演变的时空变异性尺度，并且在某种意义上通过“边缘化”公共组件来构建这种复杂的依赖性。具体而言，两个模型框架：</p>
<ul>
<li>(a) 由多个连接的伸缩级别组成；</li>
<li>(b) 包括降维阶段；</li>
<li>© 通常不对二阶依赖建模（注意，高斯过程网络和受限玻尔兹曼机是一个例外）；</li>
<li>(d) 可以处理多个输入（预测变量）和不同的输出类型；</li>
<li>(e) 有大量参数需要估计；</li>
<li>(f) 需要大量训练数据；</li>
<li>(g) 需要先验信息（或预训练、启发式等）；</li>
<li>(h) 需要正规化；</li>
<li>(i) 计算成本高，需要高效的算法实现。</li>
</ul>
<p>上述几点表明，DH-DSTM 和 DN-DSTM 框架面临的主要挑战之一与实现和计算有关。</p>
<p>也就是说，在 DH-DSTM 框架中，必须做出许多决定，涉及依赖结构的类型，是将结构放入协方差还是均值，要包含的机制信息量以及先验分布，仅举一例很少。此外，在这些复杂的建模情况下，通常必须使用某种相对高效的语言从头开始对 DH-DSTM 进行编程，因为执行贝叶斯计算的自动化程序包通常不够灵活以适应 DH-DSTM，或者效率太低（即，它们在提供通用解决方案方面的优势可能会限制某些特定的依赖结构）。同样，DN-DSTM 模型也可以有大量的调整参数和模型选择（例如，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 的选择、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span>、层数、每层隐藏单元的数量、正则化的类型、预训练等）。尽管上述参考资料包含对某些情况的建议，但对于这些决定没有普遍的建议——这在很大程度上是一种经验和反复试验的努力。然而，与 DH-DSTM 不同的是，Tensor Flow、Theano、Caffe、pyTorch（以及更多！）等标准软件环境非常灵活，并且在某种意义上是模块化的，这增加了它们在生产环境中的实用性。</p>
<p>建模范例之间还有许多其他结构差异。首先，DH-DSTM 框架基于随机模型，该模型包括有效概率构造中的分布误差项（即所有随机分量的联合分布可以写成一系列条件模型）。相比之下，DN-DSTM 框架是确定性的，没有误差项（请注意，当使用储层方法时（例如，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 的 ESN），则 <code>式(17)</code> 是随机变换而不是正式的随机模型） . DN-DSTM 缺乏概率结构的一个后果是没有明确的机制来生成基于模型的 DN-DSTM 预测或分类不确定性估计。其次，人们在对参数进行推断时受到限制——尽管应该注意的是，在这种类型的模型中，这很少会引起人们的兴趣，因为参数通常是不可识别的、高度依赖的和不可解释的。</p>
<p>此外，关于如何在深度 NN 框架中普遍包含已知关系（例如，如机制模型所建议的）仍然是一个悬而未决的问题（尽管有关该领域的近期工作，请参阅 Karpatne 等人，2017 年）。也就是说，DN-DSTM 框架确实有一些重要的优势，因为它很容易在许多现有软件包中实现的反向传播估计范例中操作和实现不同的模型结构（例如，堆叠不同的模型组件）。最后，在时空动力学的背景下，应该注意的是，RNN 结构可以自然地适应非马尔可夫动力学（例如，对遥远过去事件的记忆）。最后一点对环境、生态和农业应用具有潜在的重要意义，但并未成为时空模型统计实现的重点。</p>
<h2 id="5-结合-DH-DSTM-和-DN-DSTM-框架">5 结合 DH-DSTM 和 DN-DSTM 框架</h2>
<p>结合 DH-DSTM 和 DN-DSTM 框架的一种自然方法是允许 DN-DSTM 中的参数是随机的，或许添加一些误差项，然后通过贝叶斯范式实施。尽管至少从 1990 年代开始就考虑了神经网络的贝叶斯实现（MacKay，1992 年；Neal，1996 年），但由于大量的依赖和不可识别的依赖关系，从完全贝叶斯的角度实现深度神经模型极具挑战性参数（参见 Polson 等人，2017 年的概述）。此类模型可以在某些情况下实施（例如 Chatzis，2015 年；Chien 和 Ku，2016 年；Gan 等人，2016 年；McDermott 和 Wikle，2017a），但对特定数据集非常敏感，并且通常在计算上令人望而却步。最近，变分贝叶斯 (Tran et al., 2018) 和可扩展贝叶斯方法 (Snoek et al., 2015) 等近似贝叶斯方法已成功用于深度模型。在 DN-DSTM 的背景下，这仍然是一个活跃的研究领域。</p>
<p>或者，最近使用两种相对简单的方法来混合 DN-DSTM 和 DH-DSTM 范例。这些这样做的方式也减轻了与实施 DH-DSTM 相关的挑战。也就是说，DH-DSTM 通常在参数空间中遭受维数灾难，需要大量数据和相当专业的计算算法，因此开发和实施效率相当低。混合方法减轻了这些问题，但仍然提供了一种灵活有效的方法来以考虑不确定性量化的方式对复杂的时空过程进行建模。</p>
<h3 id="5-1-集成方法">5.1 集成方法</h3>
<p>McDermott 和 Wikle (2017b) 对标准 ESN 模型进行了多项修改，以解释时空非线性预测设置中不确定性量化的简单方法。他们考虑了二次 ESN 模型。也就是说，对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t = 1,\ldots, T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> , 让</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mi mathvariant="bold">R</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">s</mi><mi mathvariant="bold">p</mi><mi mathvariant="bold">o</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">s</mi><mi mathvariant="bold">e</mi></mrow><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold">z</mi><mi>t</mi></msub><mo>=</mo><msub><mi mathvariant="bold">V</mi><mn>1</mn></msub><msub><mi mathvariant="bold">y</mi><mi>t</mi></msub><mo>+</mo><msub><mi mathvariant="bold">V</mi><mn>2</mn></msub><msubsup><mi mathvariant="bold">y</mi><mi>t</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi mathvariant="bold-italic">ϵ</mi><mi>t</mi></msub><mo separator="true">,</mo><mspace width="1em"/><msub><mi mathvariant="bold-italic">ϵ</mi><mi>t</mi></msub><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mi mathvariant="bold">I</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mi mathvariant="bold">H</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">d</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">e</mi></mrow><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold">y</mi><mi>t</mi></msub><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mfrac><mi>ν</mi><mrow><mi mathvariant="normal">∣</mi><msub><mi>λ</mi><mi>w</mi></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mi mathvariant="bold">W</mi><msub><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi mathvariant="bold">U</mi><msub><mover accent="true"><mi mathvariant="bold">x</mi><mo>~</mo></mover><mi>t</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mi mathvariant="bold">P</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">r</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">m</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">t</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">r</mi><mi mathvariant="bold">s</mi></mrow><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="bold">W</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>w</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><msub><mo stretchy="false">]</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo>:</mo><msub><mi>w</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo>=</mo><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>w</mi></msubsup><mtext>Unif</mtext><mo stretchy="false">(</mo><mo>−</mo><msub><mi>a</mi><mi>w</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>w</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>w</mi></msubsup><mo stretchy="false">)</mo><msub><mi>δ</mi><mn>0</mn></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="bold">U</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>u</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><msub><mo stretchy="false">]</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>:</mo><msub><mi>u</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow><mi>u</mi></msubsup><mtext>Unif</mtext><mo stretchy="false">(</mo><mo>−</mo><msub><mi>a</mi><mi>u</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>u</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow><mi>u</mi></msubsup><mo stretchy="false">)</mo><msub><mi>δ</mi><mn>0</mn></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>w</mi></msubsup><mo>∼</mo><mtext>Bern</mtext><mo stretchy="false">(</mo><msub><mi>π</mi><mi>w</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow><mi>u</mi></msubsup><mo>∼</mo><mtext>Bern</mtext><mo stretchy="false">(</mo><msub><mi>π</mi><mi>u</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\mathbf{Response}: &amp;\mathbf{z}_t = \mathbf{V}_1 \mathbf{y}_t + \mathbf{V}_2 \mathbf{y}^2_t + \boldsymbol{\epsilon}_t, \quad \boldsymbol{\epsilon}_t \sim  \text{Gau}(\mathbf{0}, \sigma^2 \mathbf{I}) \tag{18}\\
\mathbf{Hidden State}: &amp;\mathbf{y}_t = g (\frac{\nu }{|λ_w|} \mathbf{W} \mathbf{y}_{t−1} + \mathbf{U} \tilde{\mathbf{x}}_t)  \tag{19}\\
\mathbf{Parameters}: &amp;\mathbf{W} = [w_{i, \ell}]_{i, \ell} : w_{i, \ell} = γ^w_{i, \ell} \text{Unif} (−a_w, a_w) + (1 − γ^w_{i, \ell}) δ_0 \tag{20}\\
&amp;\mathbf{U} = [u_{i,j} ]_{i,j} : u_{i,j} = γ^u_{i,j} \text{Unif} (−a_u, a_u) + (1 − γ^u_{i,j}) δ_0 \tag{21}\\
&amp;γ^w_{i, \ell} \sim  \text{Bern}(π_w) \tag{22}\\
&amp;γ^u_{i,j} \sim  \text{Bern}(π_u) \tag{23}
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.9601em;vertical-align:-4.7301em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.2301em;"><span style="top:-7.4735em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Response</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-5.7059em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">HiddenState</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-3.6299em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">Parameters</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-2.1068em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"></span></span><span style="top:-0.5837em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"></span></span><span style="top:0.9394em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.7301em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.2301em;"><span style="top:-7.4735em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">ϵ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">ϵ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathbf">I</span><span class="mclose">)</span></span></span><span style="top:-5.7059em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">U</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">x</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.6299em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">Unif</span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1068em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"></span><span class="mord mathbf">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">Unif</span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-0.5837em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">Bern</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:0.9394em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">Bern</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.7301em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.2301em;"><span style="top:-7.4735em;"><span class="pstrut" style="height:3.1076em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">18</span></span><span class="mord">)</span></span></span></span><span style="top:-5.7059em;"><span class="pstrut" style="height:3.1076em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">19</span></span><span class="mord">)</span></span></span></span><span style="top:-3.6299em;"><span class="pstrut" style="height:3.1076em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">20</span></span><span class="mord">)</span></span></span></span><span style="top:-2.1068em;"><span class="pstrut" style="height:3.1076em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">21</span></span><span class="mord">)</span></span></span></span><span style="top:-0.5837em;"><span class="pstrut" style="height:3.1076em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">22</span></span><span class="mord">)</span></span></span></span><span style="top:0.9394em;"><span class="pstrut" style="height:3.1076em;"></span><span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">23</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.7301em;"><span></span></span></span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo separator="true">⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(·)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span> 是一个激活函数（通常是双曲正切函数），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>w</mi></msub></mrow><annotation encoding="application/x-tex">λ_w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是“谱半径”（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> 的最大特征值），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">ν</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\nu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06898em;">ν</span></span></span></span></span></span> 是一个缩放参数，取值在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> 之间，有助于控制系统内存量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">U</mi></mrow><annotation encoding="application/x-tex">\mathbf{U}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">U</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">V</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{V}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">V</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{V}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为权重矩阵，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>σ</mi><mi>o</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_o</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为狄拉克函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>w</mi></msubsup></mrow><annotation encoding="application/x-tex">γ^w_{i, \ell}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.4192em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192em;"><span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mi>u</mi></msubsup></mrow><annotation encoding="application/x-tex">γ^u_{i, \ell}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0836em;vertical-align:-0.4192em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192em;"><span></span></span></span></span></span></span></span></span></span> 表示指示变量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><mi>w</mi></msub></mrow><annotation encoding="application/x-tex">\pi_w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">\pi_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示参数在权重矩阵为 0。注意，除以<code>式（19）</code> 中的光谱半径可确保前面提到的回波状态属性，而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span> 控制内存。在此模型中估计的唯一参数是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">V</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{V}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">V</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{V}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 中的参数，以及 <code>式 (18)</code> 中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，为此我们使用岭惩罚超参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">r_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。同样，重要的是要注意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">U</mi></mrow><annotation encoding="application/x-tex">\mathbf{U}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">U</span></span></span></span> 不是估计的，而只是分别从 <code>式(20)</code> 和 <code>式(21)</code> 中得出。超参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><mi>w</mi></msub></mrow><annotation encoding="application/x-tex">\pi_w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>π</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">\pi_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>w</mi></msub></mrow><annotation encoding="application/x-tex">a_w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">a_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">r_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的指定如下所述。</p>
<p>ESN 的修改使其可用作 DSTM，包括显式误差项 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">ϵ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\epsilon}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">ϵ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、二次项 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">V</mi><mn>2</mn></msub><msubsup><mi mathvariant="bold">y</mi><mi>t</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{V}_2 \mathbf{y}^2_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> 以及最重要的输入向量嵌入：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi mathvariant="bold">x</mi><mo>~</mo></mover><mi>t</mi></msub><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi mathvariant="bold">x</mi><mi>t</mi><mo mathvariant="normal">′</mo></msubsup><mo separator="true">,</mo><msubsup><mi mathvariant="bold">x</mi><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow><mo mathvariant="normal">′</mo></msubsup><mo separator="true">,</mo><msubsup><mi mathvariant="bold">x</mi><mrow><mi>t</mi><mo>−</mo><mn>2</mn><mi>τ</mi></mrow><mo mathvariant="normal">′</mo></msubsup><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msubsup><mi mathvariant="bold">x</mi><mrow><mi>t</mi><mo>−</mo><mi>m</mi><mi>τ</mi></mrow><mo mathvariant="normal">′</mo></msubsup><msup><mo stretchy="false">]</mo><mo mathvariant="normal">′</mo></msup><mtext>。</mtext></mrow><annotation encoding="application/x-tex">\tilde{\mathbf{x}}_t = [\mathbf{x}^\prime_t, \mathbf{x}^\prime_{t−τ} , \mathbf{x}^\prime_{t−2τ} ,\ldots , \mathbf{x}^\prime_{t−mτ}]^\prime。
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8313em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">x</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1072em;vertical-align:-0.3053em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span><span class="mord cjk_fallback">。</span></span></span></span></span></p>
<p>嵌入包括输入预测变量的滞后值，并且由于 Takens 的理论（Takens，1981）在动力系统中很重要，该理论指出可以通过足够大数量的一部分的滞后值来表示高维状态空间状态空间。请注意，结果对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>π</mi><mi>w</mi></msub><mo separator="true">,</mo><msub><mi>π</mi><mi>u</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>w</mi></msub><mo separator="true">,</mo><msub><mi>a</mi><mi>u</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pi_w, \pi_u, a_w, a_u\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 不是很敏感，它们通常固定在较小的值，但结果可能对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>n</mi><mi>h</mi></msub><mo separator="true">,</mo><mi>ν</mi><mo separator="true">,</mo><msub><mi>r</mi><mi>v</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{n_h, \nu, r_v\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 敏感，因此通过交叉验证选择它们。</p>
<p>McDermott 和 Wikle（2017b）考虑了一种简单的集合预测方法（类似于参数自举；Sheng 等人（2013）），其中从储层矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">U</mi></mrow><annotation encoding="application/x-tex">\mathbf{U}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">U</span></span></span></span> 中提取多个样本，并为每个参数集重新拟合模型。这给出了输出预测的分布，并允许量化预测中的不确定性。他们提出了一个例子，其中这个二次集合 ESN (Q-EESN) 模型用于生成热带太平洋 SST 的长期（6 个月）预报（即 El Nino 和 La Nina 事件）。该模型表现非常好。例如，<code>图 6</code> 显示了 2017 年 12 月 SST 预测的预测和预测不确定性，给定的数据截至 2017 年 6 月（展示了 La Nina事件）。</p>
<p>但是请注意，美国国家海洋和大气局气候预测中心 (CPC) 和哥伦比亚大学国际气候与社会研究所 (IRI) 对同一时期提出的动力和统计预测并未表明 La Nina 会发展（他们对这一时期的 La Nina 的概率预测约为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>15</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">15\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">15%</span></span></span></span>）。这里 Q-EESN 方法成功的原因可能与 ESN 是一个包含非线性相互作用的动态模型这一事实有关，而且它还增加了输入空间以执行回归（Gallicchio 和 Micheli，2011）。也就是说，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{y}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的维度通常大于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mspace linebreak="newline"></mspace><mi>t</mi><mi>i</mi><mi>l</mi><mi>d</mi><mi>e</mi><msub><mi mathvariant="bold">x</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\\tilde{\mathbf{x}}_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">d</span><span class="mord mathnormal">e</span><span class="mord"><span class="mord"><span class="mord mathbf">x</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>（即潜在预测变量的维度扩展）。此外，小的、稀疏的、随机的权重限制了过度拟合并规范了回归。最后，Q-EESN 实现中的嵌入式输入允许额外的非线性，并且隐藏单元相对较少的集成引导方法提供了一个“弱学习者委员会”。重要的是要注意，与传统 DH-DSTM 方法需要数小时相比，这种方法在笔记本电脑上只需几秒钟即可实现。</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20221206194306-f80a.webp" alt="Figure05"></p>
<blockquote>
<p>图 5：左图：根据 2017 年 12 月到 2017 年 6 月的观测，太平洋 SST 的 6 个月预测的长期预测摘要图。第一行显示观测到的 SST 异常（与气候平均值的偏差）。 Q-EESN 模型的预测平均值显示在第二个子图中，底部两个子图显示每个网格单元中计算的 95% 预测区间的下分位数和上分位数。右图显示了所谓的 Nino3.4 指数的 Q-EESN 预测分布，该指数基于左侧第二个子图中框表示的区域的平均值。蓝星表示 Q-EESN 预测平均值，观测到的指数值由实心蓝色圆圈表示。实心和空心红色圆圈对应于 IRI/CPC 提供的确定性和随机模型基于相同起始和验证期的预测（请参阅网站文本中的脚注）。</p>
</blockquote>
<h3 id="5-2-深度基函数方法">5.2 深度基函数方法</h3>
<p>Q-EESN 模型没有链接隐藏层的机制，这对于在多个时间尺度上发生的过程很重要。机器学习文献中已经实施了深度 ESN 模型（例如，Jaeger，2007 年；Triefenbach 等人，2013 年；Antonelo 等人，2017 年；Ma 等人，2017 年；Gallicchio 等人，2018 年），但是这些方法通常不适应不确定性量化，也不是为时空系统设计的。然而，可以扩展这些深度 ESN 模型以适应 <code>式(16)</code> 中的时空过程。例如，McDermott 和 Wikle（2018 年）在一个集合参数引导上下文中这样做，以解释多个时间尺度和预测中的不确定性。他们还考虑了一种实现，其中 <code>式(16)</code> 用于生成作为输入的随机变换的基函数。这在时空回归上下文中特别有用，即当人们试图根据另一个时空过程来预测一个时空过程时。具体来说，考虑模型：</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Data Stage</mtext><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold">z</mi><mi>t</mi></msub><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mi mathvariant="bold">Φ</mi><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">C</mi><mi>z</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Output Stage</mtext><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold-italic">α</mi><mi>t</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow></msub></munderover><mrow><mo fence="true">[</mo><msubsup><mi mathvariant="bold-italic">β</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><munderover><mo>∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>2</mn></mrow><mi>L</mi></munderover><msubsup><mi mathvariant="bold-italic">β</mi><mi mathvariant="normal">ℓ</mi><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><mtext> </mtext><msubsup><mover accent="true"><mi mathvariant="bold">y</mi><mo>~</mo></mover><mrow><mi>t</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub><mo separator="true">,</mo><mspace width="1em"/><msub><mi mathvariant="bold-italic">η</mi><mi>t</mi></msub><mo>∼</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mn mathvariant="bold">0</mn><mo separator="true">,</mo><msubsup><mi>σ</mi><mi>η</mi><mn>2</mn></msubsup><mi mathvariant="bold">I</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Priors</mtext><mo>:</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msubsup><mi>β</mi><mrow><mi mathvariant="normal">ℓ</mi><mo separator="true">,</mo><mi>b</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup><mi mathvariant="normal">∣</mi><msubsup><mi>γ</mi><mi mathvariant="normal">ℓ</mi><msub><mi>β</mi><mi mathvariant="normal">ℓ</mi></msub></msubsup><mo>∼</mo><msubsup><mi>γ</mi><mi mathvariant="normal">ℓ</mi><msub><mi>β</mi><mi mathvariant="normal">ℓ</mi></msub></msubsup><mtext>Gau</mtext><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><msubsup><mi>σ</mi><mrow><msub><mi>β</mi><mi mathvariant="normal">ℓ</mi></msub><mo separator="true">,</mo><mn>0</mn></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>γ</mi><mi mathvariant="normal">ℓ</mi><msub><mi>β</mi><mi mathvariant="normal">ℓ</mi></msub></msubsup><mo stretchy="false">)</mo><mtext>Gau</mtext><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><msubsup><mi>σ</mi><mrow><msub><mi>β</mi><mi mathvariant="normal">ℓ</mi></msub><mo separator="true">,</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msubsup><mi>γ</mi><mi mathvariant="normal">ℓ</mi><msub><mi>β</mi><mi mathvariant="normal">ℓ</mi></msub></msubsup><mo>∼</mo><mtext>Bernoulli</mtext><mo stretchy="false">(</mo><msub><mi>π</mi><mrow><mi>β</mi><mi mathvariant="normal">ℓ</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msubsup><mi>σ</mi><mi>η</mi><mn>2</mn></msubsup><mo>∼</mo><mtext>IG</mtext><mo stretchy="false">(</mo><msub><mi>α</mi><mi>η</mi></msub><mo separator="true">,</mo><msub><mi>β</mi><mi>η</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}
\text{Data Stage}: &amp;\mathbf{z}_t \sim  \text{Gau}(\boldsymbol{\Phi} \boldsymbol{\alpha}_t, \mathbf{C}_z) \\
\text{Output Stage}: &amp;\boldsymbol{\alpha}_{t} = \sum^{n_{res}}_{j=1} \left[ \boldsymbol{\beta}^{(j)}_1 \mathbf{y}^{(j)}_{t,1} +  \sum^L_{\ell=2} \boldsymbol{\beta}^{(j)}_\ell \tilde{\mathbf{y}}^{(j)}_{t, \ell} \right] + \boldsymbol{\eta}_t, \quad \boldsymbol{\eta}_t \sim  \text{Gau}(\mathbf{0}, \sigma^2_η \mathbf{I}) \\
\text{Priors}: &amp;\beta^{(j)}_{\ell,b} | γ^{\beta_\ell}_{\ell} \sim  γ^{\beta_\ell}_{\ell} \text{Gau}(0, \sigma^2_{\beta_\ell,0}) + (1 − γ^{\beta_\ell}_\ell ) \text{Gau}(0, \sigma^2_{\beta_\ell,1}), \\
&amp;γ^{\beta_\ell}_\ell \sim  \text{Bernoulli}(\pi_{\beta \ell}) \\
&amp;\sigma^2_η \sim  \text{IG}(\alpha_η, \beta_η)
\end{align*}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.9986em;vertical-align:-4.7493em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.2493em;"><span style="top:-8.2376em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Data Stage</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-5.7493em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Output Stage</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-2.9907em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord text"><span class="mord">Priors</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span><span style="top:-1.2863em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"></span></span><span style="top:0.2378em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.7493em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.2493em;"><span style="top:-8.2376em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathbf">Φ</span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-5.7493em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">α</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6625em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3111em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">res</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4337em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4337em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"> </span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4374em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">η</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1864em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord mathbf">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord mathbf">I</span><span class="mclose">)</span></span></span><span style="top:-2.9907em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4374em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.3987em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0528em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.3987em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0528em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0528em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.3987em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0528em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord text"><span class="mord">Gau</span></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0528em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span><span style="top:-1.2863em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.3987em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0528em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">ℓ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">Bernoulli</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:0.2378em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">IG</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">η</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.7493em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{y}^{(j)}_{t,1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4472em;vertical-align:-0.4024em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4337em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4024em;"><span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">y</mi><mrow><mi>t</mi><mo separator="true">,</mo><mi mathvariant="normal">ℓ</mi></mrow><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{y}^{(j)}_{t, \ell}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4822em;vertical-align:-0.4374em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mpunct mtight">,</span><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4374em;"><span></span></span></span></span></span></span></span></span></span> 是 <code>式(16)</code> 中给出的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi mathvariant="bold">x</mi><mo>~</mo></mover><mrow><mi>t</mi><mo>−</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\tilde{\mathbf{x}}_{t-\tau}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8896em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6813em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">x</span></span><span style="top:-3.3634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold-italic">β</mi><mi mathvariant="normal">ℓ</mi><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\boldsymbol{\beta}^{(j)}_\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3461em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03403em;">β</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">ℓ</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span> 是第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 个系综和第  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span> 个的相关回归系数等级。重要的是，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">y</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{y}_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是从集成深度 ESN “离线” 生成的，具有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span> 的主成分缩减阶段。此外，</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>π</mi><mrow><mi>w</mi><mn>1</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>π</mi><mrow><mi>w</mi><mi>L</mi></mrow></msub><mo separator="true">,</mo><msub><mi>π</mi><mrow><mi>u</mi><mn>1</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>π</mi><mrow><mi>u</mi><mi>L</mi></mrow></msub><mo separator="true">,</mo><msub><mi>a</mi><mrow><mi>w</mi><mn>1</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mrow><mi>w</mi><mi>L</mi></mrow></msub><mo separator="true">,</mo><msub><mi>a</mi><mrow><mi>u</mi><mn>1</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mrow><mi>u</mi><mi>L</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pi_{w1}, \ldots, \pi_{wL}, \pi_{u1} , \ldots , \pi_{uL} , a_{w1} , \ldots, a_{wL} , a_{u1} ,\ldots , a_{uL} \}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">uL</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">uL</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span></span></p>
<p>固定为较小的值，并且除了第一层之外的所有层的隐藏单元数都是固定的，因为所有这些层都经过降维函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal">Q</span></span></span></span>。最后，</p>
<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ν</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>ν</mi><mi>L</mi></msub><mo separator="true">,</mo><msub><mi>n</mi><mrow><mover accent="true"><mi>h</mi><mo>~</mo></mover><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>n</mi><mrow><mover accent="true"><mi>h</mi><mo>~</mo></mover><mo separator="true">,</mo><mi>L</mi></mrow></msub><mo separator="true">,</mo><msub><mi>n</mi><mrow><mi>h</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>r</mi><mi>ν</mi></msub><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\nu_1, \ldots , \nu_L, n_{\tilde{h},2}, \ldots , n_{\tilde{h} ,L}, n_{h,1}, r_\nu, m \}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1932em;vertical-align:-0.4432em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3929em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mathnormal mtight">h</span></span><span style="top:-3.3134em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">~</span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4432em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3929em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mathnormal mtight">h</span></span><span style="top:-3.3134em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord mtight">~</span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4432em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">h</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">}</span></span></span></span></span></p>
<p>由遗传算法选择。参数引导方法生成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">j = 1, \ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span> , 与上面的 Q-EESN 模型 <code>式(21)</code> 和 <code>式(20)</code> 一样，通过对不同的权重矩阵进行采样来获取这些深度 ESN 的集合。</p>
<p>例如，McDermott 和 Wikle（2018 年）考虑了太平洋海温下美国玉米带土壤湿度的 6 个月长期预报。<code>图 6</code> 显示了基于 3 级深度集合 ESN 模型的 2014 年 5 月样本外预测，给定 2017 年 11 月的 SST。他们表明，与各种模型相比，该模型在连续排名概率得分方面表现最佳，在均方预测误差方面表现第二（该模型的 2 级版本在该指标上表现略好）。</p>
<p>这种方法本质上是一个高维回归问题，其中通过深度 ESN 模型对输入进行随机变换来生成一组基函数。多个这样的转换被认为是潜在的预测因素，以赋予方法灵活性和可重复性。大量预测变量由 SSVS 正则化控制。请注意，此模型中的输入（预测变量）是随机和动态转换的。因此，时空回归模型本身不是动态的，但重要的是，转换通过 ESN 结构是动态的。这些多级转换允许预测变量中的不同时间和空间尺度影响响应。重要的是，通过在转换（离线）中包含动态，该框架非常容易通过正则化回归方法实现，并且由于 ESN 中的储层方法和相对有效（与深度参数统计模型和深度机器学习模型相比）简单的正则化。这里的数据模型可以轻松适应其他数据类型，例如广义线性混合模型的深度贝叶斯实现（例如，Tran 等人，2018 年）。</p>
<p><img src="https://xishansnowblog.oss-cn-beijing.aliyuncs.com/images/images/stats-20221206195054-c34a.webp" alt="Figure06"></p>
<blockquote>
<p>图 6：使用 3 层贝叶斯深度集合回波状态网络模型对 2014 年 5 月土壤水分长期预测的后验总结。 (a) 每个空间网格位置的观测土壤水分值。 (b) 每个网格位置的后验预测平均值。 © 每个网格位置的后验预测标准差。每个图都通过各自的方法和标准偏差进行了标准化以帮助可视化，并且为了可视化而移除了极端异常值（由黑色方格表示）。有关详细信息，请参阅 McDermott 和 Wikle (2018)。</p>
</blockquote>
<h2 id="6-讨论">6 讨论</h2>
<p>DH-DSTM 的基本原则之一是，要对跨多个时间和空间尺度的复杂过程进行建模，可以从考虑一系列关联的概率模型中获益。特别是，由于很难为复杂（例如，非线性）时空过程指定依赖结构，因此人们将建模工作置于条件均值中，并通过边缘化利用建立依赖性。同样，在过去十年中在图像和语言处理领域变得如此流行的机器学习深度神经模型（例如 DNN、CNN、RNN）也基于一系列链接模型（通常不是随机模型），其中一个层次的输出成为下一个层次的输入。这些模型的时空版本 DN-DSTM 通常结合 CNN 和 RNN，并且还试图通过了解哪些空间和/或时间变异性尺度对于预测响应很重要来构建复杂性。这些建模框架有许多共同的实际问题，包括需要大型训练数据集、降维、正则化和高效计算。最近缓解其中一些问题的方法，例如，在没有大量训练数据时应用模型，受益于在 ESN 的背景下考虑水库计算。在时空问题中，这些模型通过使用参数自举和基函数变换方法被置于统计环境中。这些可以以传统 DH-DSTM 的一小部分成本实现，但仍保留概率公式以允许不确定性量化，并受益于 DN-DSTM 灵活地模拟多个时间和空间尺度的能力。</p>
<p>在混合用于环境、生态和环境统计的 DH-DSTM 和 DN-DSTM 方面，我们只是触及了皮毛。一个重要的挑战是能够在此混合框架中有效地包含机制信息。传统上，由于机制公式和灵活学习公式之间的冲突，以及通过基于梯度的优化训练此类模型的挑战，将此类信息包含在 DN-DSTM 中一直具有挑战性。此外，通过包含深度强化学习的想法可以获得潜在的进步（例如，参见 Aggarwal，2018 年的概述）。这些方法训练模型的方式是，它们会因做出好的决定而获得奖励，并因做出糟糕的决定而受到惩罚。这是用于 AlphaGo（Silver 等人，2016 年）和后来的游戏算法（Silver 等人，2018 年）的技术。考虑到在控制工程中使用强化学习的悠久历史，在环境统计中与 DH-DSTMs 的有用联系是可能的。此外，DH-DSTM 和 DN-DSTM 的混合可能会受益于生成对抗网络的最新进展（Goodfellow 等人，2014 年）。这种方法以受益于两个相互竞争的 NN 的方式训练模型。特别是，一个网络生成潜在的解决方案，另一个网络评估或区分这些解决方案。事实上，深度神经建模方面的文献发展非常迅速，很高兴看到这些方法中的哪些可以包含在更传统的概率 DSTM 框架中。</p>
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</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">文章作者: </span><span class="post-copyright-info"><a href="http://xishansnow.github.io">西山晴雪</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">文章链接: </span><span class="post-copyright-info"><a href="http://xishansnow.github.io/posts/4c2ac315.html">http://xishansnow.github.io/posts/4c2ac315.html</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="http://xishansnow.github.io" target="_blank">西山晴雪的知识笔记</a>！</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/GeoAI/">GeoAI</a><a class="post-meta__tags" href="/tags/%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0/">深度学习</a><a 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id="aside-content"><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span><span class="toc-percentage"></span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#1-%E7%AE%80%E4%BB%8B"><span class="toc-text">1 简介</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-%E6%97%B6%E7%A9%BA%E5%BB%BA%E6%A8%A1%E6%A6%82%E8%BF%B0"><span class="toc-text">2 时空建模概述</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#2-1-%E5%8A%A8%E6%80%81%E6%97%B6%E7%A9%BA%E6%A8%A1%E5%9E%8B%EF%BC%88DSTM%EF%BC%89"><span class="toc-text">2.1 动态时空模型（DSTM）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-2-%E5%9F%BA%E5%87%BD%E6%95%B0%E8%A1%A8%E7%A4%BA"><span class="toc-text">2.2 基函数表示</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#3-%E6%B7%B1%E5%B1%82%E5%88%86%E5%B1%82%E7%BB%9F%E8%AE%A1%E6%A8%A1%E5%9E%8B"><span class="toc-text">3 深层分层统计模型</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#3-1-%E6%B7%B1%E5%BA%A6%E5%88%86%E5%B1%82%E5%8A%A8%E6%80%81%E6%97%B6%E7%A9%BA%E6%A8%A1%E5%9E%8B%EF%BC%88DH-DSTMs%EF%BC%89"><span class="toc-text">3.1 深度分层动态时空模型（DH-DSTMs）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#3-2-DH-DTSM-%E7%A4%BA%E4%BE%8B%EF%BC%9A%E6%B5%B7%E6%B4%8B%E6%B0%B4%E8%89%B2"><span class="toc-text">3.2 DH-DTSM 示例：海洋水色</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#4-%E6%B7%B1%E5%BA%A6%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C%E6%A8%A1%E5%9E%8B"><span class="toc-text">4 深度神经网络模型</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#4-1-%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C"><span class="toc-text">4.1 神经网络</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-2-%E6%B7%B1%E5%BA%A6%E5%89%8D%E9%A6%88%E7%BD%91%E7%BB%9C-DNN"><span class="toc-text">4.2 深度前馈网络 (DNN)</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-3-%E5%8D%B7%E7%A7%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C%EF%BC%88CNN%EF%BC%89"><span class="toc-text">4.3 卷积神经网络（CNN）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-4-%E5%BE%AA%E7%8E%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C%EF%BC%88RNNs%EF%BC%89"><span class="toc-text">4.4 循环神经网络（RNNs）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-5-%E5%9B%9E%E6%B3%A2%E7%8A%B6%E6%80%81%E7%BD%91%E7%BB%9C%EF%BC%88ESN%EF%BC%89"><span class="toc-text">4.5 回波状态网络（ESN）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-6-%E6%B7%B1%E5%BA%A6%E7%A5%9E%E7%BB%8F-DSTM%EF%BC%88DN-DSTM%EF%BC%89"><span class="toc-text">4.6 深度神经 DSTM（DN-DSTM）</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#4-7-DH-DSTMs%E5%92%8CDN-DSTMs%E4%B9%8B%E9%97%B4%E7%9A%84%E8%BF%9E%E6%8E%A5"><span class="toc-text">4.7 DH-DSTMs和DN-DSTMs之间的连接</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#5-%E7%BB%93%E5%90%88-DH-DSTM-%E5%92%8C-DN-DSTM-%E6%A1%86%E6%9E%B6"><span class="toc-text">5 结合 DH-DSTM 和 DN-DSTM 框架</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#5-1-%E9%9B%86%E6%88%90%E6%96%B9%E6%B3%95"><span class="toc-text">5.1 集成方法</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#5-2-%E6%B7%B1%E5%BA%A6%E5%9F%BA%E5%87%BD%E6%95%B0%E6%96%B9%E6%B3%95"><span class="toc-text">5.2 深度基函数方法</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#6-%E8%AE%A8%E8%AE%BA"><span class="toc-text">6 讨论</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE"><span 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